Red Dawn (1984): A classic example, in which the USSR invades the Western United States and a group of high schoolers' efforts to stop it. The remake has North Korean invaders (originally Chinese, but changed in order to have it marketable there).
Conversed in Hot Tub Time Machine. Blaine's gang thinks that the time traveling main characters are actually Soviet spies, due to their odd behavior, and the modern-day gadgets that he found in their bags. He mentions the film Red Dawn (1984), which he owns a poster of in his room.
Discussed in Pearl Harbor during Roosevelt's first meeting with his advisers after the namesake attack. One general, in underlining just how poor a state the US military is in, states that if the Japanese invaded at that moment they would get as far east as Chicago.
The Man in the High Castle: Set after the fact, this book depicts an alternate 1960s in which the US was taken over by Japan and Germany from the West and East coasts respectively. The Rocky Mountain States is a Japanese puppet.
The Fall Of A Nation by Thomas Dixon (author of The Klansman, aka The Birth of a Nation). Written prior to US involvement in World War One, it has the United States refusing to intervene in the Great War, which stalls to a stalemate. Peace is declared, but it's actually a ruse to allow the combined armies of Europe to invade America.
The Mouse That Roared: The Duchy of Grand Fenwick declares war on the U.S. with the intent of losing immediately and raking in the reparation money. So they send over a small army in Medieval armor and spears, expecting to surrender upon reaching American soil. It doesn't quite work out that way.
Robert Conroy's Alternate History novel 1901 concerns a relatively limited invasion of the Northeast in the title year by the forces of Kaiser Wilhelm II. The Kaiser's objective here is not to conquer the United States outright, but to force the U.S. to turn over its newly-won colonial possessions of Puerto Rico and the Philippines by essentially holding New York City and surrounding areas for ransom. He also intended to confiscate the U.S. Navy, which America wouldn't need once it had no overseas territory. Unfortunately (for the Germans), the Kaiser didn't reckon with Theodore Roosevelt...
Lightning In The Night, a novel by Fred Allhoff originally serialized in Liberty magazine during 1940, recounts, in pulp-magazine style, an invasion of the United States by a grand alliance of Nazi Germany, Imperial Japan and the Soviet Union circa 1945.
Eric L Harry's Invasion has Chine invading the good old USA thanks to US disarmament. Apparently the blinkered government continued to draw down the US military despite the Chinese conquering the world by going West (having nuked Israel and beseiged Europe). That's quite an example of Ostrich thinking there!
The apocalyptic novel Resurrection Day is set in a USA where the Cuban Missile Crisis led to a nuclear exchange. With the Soviet Union destroyed, Communism fallen, and the USA crippled, the power vacuum leads to a resurgent Britain reversing its post-imperial decline - blamed on American interference - and plotting to avenge the whole 1776 business by grabbing back its North American colonies. And then some. The "Resurrection" of the title is Britain's rebirth as an undisputed world superpower, if only by default as the only country not touched by nuclear holocaust.
Star Trek: Enterprise: Several episodes featured an alternate 1940s in which the Nazis, equipped with alien technology, were able to capture portions of the East Coast.
The second episode of Sliders featured a Soviet-occupied USA.
Another episode showed a Mexican-occupied California, after the US lost the Mexican-American War.
A sort-of example: One episode features a world where the US lost the Revolution and never existed, so the country is all part of England.
Yet another episode briefly features a French-controlled USA. In this reality, Napoleon won over in Europe, never sold Louisiana to the US, and eventually conquered the rest of America.
The miniseries Amerika shows a Soviet-occupied US ten years after the invasion. Subverted though in that the US surrendered without much of a fight after an EMP pulse took out most communications. Most Americans are more interested in getting on with life than resisting.
Amerika was a work that was developed as an apology to conservatives, particularly the Reagan Administration, The Day After. The latter showed how terrible a nuclear war would be for the (many) survivors; the former argued how a Soviet victory would also be unacceptable.
Parodied by Saturday Night Live as "Amerida", a vision of what America would be like after a Canadian invasion and takeover.
The Price of Freedom RPG from West End Games, inspired by the film Red Dawn (1984). After the Soviets develop an orbital defense system that neutralizes America's strategic nuclear weapons, Soviet occupation forces enter and take over the U.S. The PCs are American freedom fighters who wage a guerrilla war against the invaders.
Invasion America (1976) board wargame by SPI (Simulations Publications Inc.). The U.S. is invaded by three international coalitions: the European Socialist Coalition, the South American Union and the Pan Asiatic League. The popularity of this game spawned a companion piece the next year, Objective: Moscow, detailing invasions of the Soviet Union by various coalitions of the Western Allies and China from the 1970's to the end of the 20th Century.
Fortress America (1986), a Milton Bradley board game. The U.S. is invaded by three international coalitions: the Euro-Socialist Pact, the Central American Federation and the Asian Peoples Alliance. Hey, wait a minute...
The box cover for the first edition of this game contains a portrait of none other than Saddam Hussein (changed for later editions). This edition of the game is now a collector's item. Also, there are some significant differences between Invasion: America and Fortress America; among other things, the former game covers the whole North American continent from the Arctic to the Panama Canal, done in typical hexagonal-map-and-paper-counter SPI/Avalon Hill, and the MB game covers only the continental United States and is done in the general style of Axis And Allies.
Shadowrun. After the U.S. was split up into the Divided States of America, Aztlan invaded both the southern part of California Free State and the Confederated American States.
The Freedom City setting for Mutants & Masterminds includes the "Erde" setting where the Axis powers won World War II and used a combination of brain-powered war machines, nuclear weapons, and mystic cataclysms to shatter the backbone of the U.S. resistance. The Time of Crisis adventure involves the heroes helping La Résistance to neutralize the war machines in the process of fighting a cosmic threat.
SS Amerika by 3W Games presents an invasion of North America (like Invasion: America, covering the entire continent) by various Axis forces during the World War II era. This is, aside from Objective: Moscow, the largest and most detailed tabletop game of this particular subgenre (four full-sized standard wargame maps), with units at the division level including the entire U.S. Army and Marine Corps WWII order of battle, plus British Commonwealth, German, Italian and Japanese units, and even contingents from various Latin American nations which can come in on either the Allied or Axis side depending on various in-game events or scenario preconditions.
Deadlands: In 1877, the British invade from Canada and capture Detroit in retaliation for American military adventurism along the Canadian border.
Axis & Allies is a World War II simulation. Invading the US is possible but pretty unlikely (it's usually easier for the Axis to get the two-capitals win condition by taking out Britain and the Soviet Union).
The World War II board game World in Flames has a variant where America is invaded by a victorious Germany called, appropriately, America in Flames.
Battlefield series: The Armored Fury booster pack for Battlefield 2 depicted maps set in Alaska and Pennsylvania as US forces defended their homeland against Chinese and Middle Eastern Collation Forces. Bad Company 2 has several multiplayer maps set in Alaska, and the ending depicts Russian forces advancing on the northern border of the US. Averted in Battlefield 3, as the levels that take place in New York are about stopping a terrorist attack, not an invasion.
Modern Warfare: The second game in the series has several levels set in Northern Virgina and D.C. The invaders are not the actual Soviet Union, since the game takes place long after the end of the Cold War, but the Ultranationalist party which has taken over Russia in the game's timeline is a pretty effective substitute. The third game has them fighting in New York City as well, before finally being pushed off US soil after their defeat there.
Appropriately, the first mission set in the invaded U.S. is called Wolverines!, and the achievement for beating it on Veteran is Red Dawn.
Call of Duty: Black Ops 2 plays with this— the US is invaded...by its own unmanned combat drones, which have been hacked to serve terrorist leader Raul Menendez's bidding. They manage to get boots on the ground as well, in the form of his mercenary army; LA is successfully laid waste to, and they make it as far as New York City and DC before Menendez self-destructs the drones to leave America defenceless.
This is also part of the plot for Call of Duty: Ghosts: The bad guys (in this case a league of hostile South American governments) have hijacked an American superweapon system that is turned on the US, devastating the country and leaving it ripe to a conventional invasion.
Homefront: The basic plot of the game is the United States under occupation by a resurgent North Korea.
World in Conflict: Set during an alternate 1980s, in which the USSR launched an invasion of the US instead of collapsing as in Real Life.
Kind of. The main battles of the war are occurring in Western Europe and the invasion force that lands in Seattle is just there to keep the USA from reinforcing its NATO allies.
Turning Point: Fall of Liberty: Set in an alternate 1950s in which the US stayed neutral in World War II and in which Churchill was killed in an accident prior to the war, causing Britain to fall to Germany, which in effect discourages the United States from ever entering the war, solidifying their stance as a neutral country. Eventually, however, the Nazis invade anyways.
End War: Battles can take place in the United States.
Rise of Nations: In the Cold War campaign, besides starting a nuclear war with them, the Soviets can also stage a conventional invasion of the United States.
Shattered Union: The European Union sends peacekeeping forces to the Washington DC area to secure international interests, whereas the Russian Federation invades and annexes Alaska during the Second American Civil War.
Took place in Fallout backstory just prior to the nuclear war, when the Chinese forces invaded Anchorage. It was eventually repulsed. The Fallout 3 expansion pack Operation Anchorage takes the player character through a VR simulation of the Anchorage Reclamation.
Descendants from a beached Chinese submarine crew can be found around what is left of San Francisco.
Ghoulified Chinese infiltrators can be found in a factory in Washington D.C. in 3.
Point Lookout has a wrecked Chinese spy submarine as part of one of its sidequests.
Red Alert 2 has an ostensibly puppet Soviet government invade the United States with the aid of psychic powers to neutralize the American nuclear arsenal.
Red Alert 3 is more global, but both the Imperial and Soviet campaigns feature invasions of the United States at some point. Going after President Ackerman at Mt. Rushmore in the Allied campaign after he goes rogue may or may not count.
Eventual end result of the Soviets developing the atomic bomb first and dropping it on Berlin in the back story of Freedom Fighters.
One trailer for Steel Battalion: Heavy Armor depicts an American invasion of Manhattan in 2080. The opposing force is implied to be related to the United Nations in some way.
To date, there have been four attempted invasions of the United States:
The first and more successful series was executed by the Royal Marines during the War of 1812, who torched all the Government buildings in Washington, DC. There were also other raids through Maine, New York state, and New Orleans, all of which were in response to the United States' invasion and attempted annexation of British North America (Canada) and the US Army's razing of Toronto when this didn't work out.
The second was by the seccessionist Confederate States of America, when Robert E. Lee led his army into Maryland (in 1862, stopped at Antietam) and Pennsylvania (in 1863, stopped at Gettysburg) (and/or 'the Northern [liberal-atheist-socialist socio-economic]invasion of the South', which some claim is still ongoing).
The third and least significant time was during The Mexican Revolution, when Pancho Villa led a raid on the border town of Columbus, New Mexico.
The fourth and second-least important was by the Imperial Marines of Imperial Japan during World War II, who seized some of the Aleutian Islands in Alaska as a part of their strategy for defeating the US Navy. Other territories like Guam and the American-run Philippines had been occupied for several months by that point, and since Alaska wasn't a state yet it's debatable whether this example even counts as an invasion of the United States.
It should be noted that none of these were an attempt to conquer American territory.
The British weren't focused on the US in 1812 because they had Napoleonic France to deal with and considered (rightly) the war an irritating sideshow.
The South was where nearly all the fighting of the US's Civil War was taking place, and General Lee calculated that if his forces could go on the offensive and defeat the Federal Government's armies in a decisive battle (before they had time to bring their massive resource-superiority to bear), they would push for a peace settlement and acknowledge the Confederation's secession. Lee's move was also motivated by a chronic food supply problem - Southern plantation holders were so hung up on growing cotton that there wasn't enough fertile land set aside for food crops; agriculture in the North, meanwhile, was mainly foodstuffs like wheat and corn.
Pancho Villa's army was so starved of supplies that he was willing to trespass into the US to get them in what was effectively a glorified supply-run.
The occupation of the Aleutians was done to divert attention from the Imperial Navy's real objective, the US Naval base at Midway (it failed in this because the US's intelligence corps had cracked the Imperial Navy's radio codes/cyphers and knew exactly what was going to happen). It was also an example of Executive Meddling, since Admiral Yamamoto's plan for Midway wasn't to capture the base but to draw the American aircraft carriers into a trap where they could be sunk (the cracked Japanese codes meant that the US Navy was instead able to reverse the trap). A diversionary operation in the Aleutians contradicted the entire point of the plan, but it happened anyway just because the higher-ups thought it was a good idea.
Nazi Germany (or more specifically, Hitler) was also gaming for this goal in the long-term. And in the even longer-term betrayal of their ally Japan, going for outright world conquest. The attempted take-over of Soviet resources during the war was to partly feed the future construction of a massive Atlantic surface fleet.
The Nazis did manage to invade part of North America when they set up monitoring stations in Greenland after seizing Denmark. However, the threat of invading Canada and the United States never came to fruition due to the actions of the Sledge Patrol (A group on 15 Greenlanders on dogsleds) who shut down the monitoring stations. Although the closest the Nazis ever got to shooting at the American mainland was U-boats harassing USN shipping in American waters.
Or so Greenlanders like to claim. There's no way Germany was ever going to win against Soviet Russia given their 'massive' materiel disadvantage, but even if they had (a huge if) then Germany would have just poured more men and resources into Greenland-based monitoring stations, making that action a mere setback.
According to a 1987 Department of Defense plan that projected the path of a Soviet invasion in case of World War III Alaska would have been invaded from across the Bering Sea in order to draw American forces away from Japan and the rest of Asia.
In 1917, Germany offered Venustiano Carranza aid to invade the U.S.A. with the promise that Mexico would take back the land that was taken in 1848 after the Mexican-American War (read: half of Mexico). This wasn't the case because by then there was this little something called The Mexican Revolution. It didn't help that the British leaked the telegram to the U.S.A.
The other problem was sheer logistics - Mexico was quite outgunned compared with the US (especially if the latter mobilized its resources for war, which an attack would certainly cause), and the only power that would have capable of giving Mexico the arms and aid needed to even consider invasion of the US was the US itself (anything from Europe would have been cut off by the US Navy).
Defense Scheme No. 1 was a Canadian plan formed in 1921 to invade the United States if there was evidence a US invasion of Canada was imminent. It would have called for four avenues of attack (the Pacific Northwest, Minnesota, Upstate New York, and Maine) in an attempt to throw US forces off balance (given the large numerical advantage the US would have) before pulling back and destroying bridges, buying time for reinforcements from Britain to arrive. Amusingly, those reinforcements would never arrive in such an event - the Royal Navy at the time believed that in a hypothetical war with the US, Canada would be impossible to defend and so would not have sent significant reinforcements. The plan (as well as the corresponding American War Plan Red formed in 1930) was largely academic in nature, though War Plan Red's declassification in 1974 caused a stir in US-Canadian relations.
An invasion of the US as things stand now at the time of typing would be very difficult, as the US has the second-largest military in the world, with 1.5 million active personnel, including the largest air force and navy (fairly important when its main rivals are overseas), and spends more on its military than most of the planet's other major nation-states combined (and that's after budget cuts; less than a decade ago the United States spent more on its military than all other nations combined). What's more, the country's liberal gun laws would ensure no end of well-armed partisans to trouble an occupying force.
During World War II, Admiral Yamamoto of Imperial Japan knew war with the US was going to end in disaster because of the country's vastly greater economic strength - the USA's GDP being ten times that of the Empire's. A couple of quotes from him on the issue:
"You cannot invade the mainland United States. There would be a rifle behind each blade of grass." (Note that this quote may be a case of Beam Me Up, Scotty!, as there is no record of him saying it, but regardless, it still makes a good point.)
"Should hostilities once break out between Japan and the United States, it is not enough that we take Guam and the Philippines, nor even Hawaii and San Francisco. We would have to march into Washington and sign the treaty in the White House. I wonder if our politicians (who speak so lightly of a Japanese-American war) have confidence as to the outcome and are prepared to make the necessary sacrifices?"
Unfortunately, the quote above was taken as a proclamation that he would do this trope instead of as a warning that this wouldn't end well.
Along with much of the above, there's also the fact that the United States is a vast, mostly urbanized, geographically diverse country flanked by two major oceans and superior interior lines. The country's sheer size would work against an invading force, though the country's highly-developed infrastructure could be made to work for them as well as for a defending force. Invading the continental United States would be slightly more forgiving than invading Russia. Slightly.
It can probably best be summed up that Invading the US has all the problems of invading Russia and/or China (Huge size, large population), with the additional difficulty of supporting said invasion across 3000 (Atlantic) or 6000 (Pacific) miles of Ocean. No nation on Earth has anywhere near the seaborne transport capacity to bring a sufficiently large army across an ocean to invade the United States, even ignoring the fact that they'd need to keep those ships from being sunk by the US Navy and Air Force.
And, to top it all off, the United States is not only a NATO member, has a number of unilateral defense treaties with other nations, and is in general much more popular than China, India, or Russia, which are the only three countries that could fend off a conventional land invasion by the USA (though not a determined naval and air campaign). Anyone looking to resist an attack by the USA would have few, if any, allies to call upon. Anyone looking to invade could well see much of the developed world in the USA's side. For now, an invasion of the USA is a 'pipe dream'.
The United States is also economically intertwined with most of the rest of the world. Any potential invader knows that, for example, cutting off American agriculture to the rest of the world will likely result in food riots, quite possibly within their own countries. China, especially, would have to worry about what would become of their economy without the U.S. to sell to, meaning they'd be far more likely to enter the war on the side of the U.S. than to be one of the invaders.
This is, however, one of the reasons China has been so active in developing relations with other countries like in Africa: other markets to sell their products to means less reliance on the US buying their stuff, which means more flexibility in their foreign policy. China might not attempt to invade Hawaii even then due to the myriad number of other obstacles, but they may be more willing to take a more firm stance in other issues like, say, Taiwan.
Lastly, none of the countries commonly depicted as invading are really interested in invading the USA as all planners view the logistics of it as sheer impossibility. In the Cold War, both the USA and the USSR dismissed any serious idea of invading and seizing the other's territory as thoroughly impractical, something which still holds today. They may have drawn up some plans as a thought exercise (a military planner has to come up with plans for damn near anything, even ones impossible to pull off), but never really thought they could do it without bringing forth The End of the World as We Know It from all the nukes it would send flying.