For much of its existence, Mexico has been a Federal Republic in much the same way China is a "People's Republic". Functionally, the government is federalist with a strong central government but states have near complete autonomy and discretionality (this last is a contested sore spot). From bottom to top: Municipal government is handled with mayoral elections and síndicos, or councilmen. This level of government usually shows the a good thermometer for national party preferences. On the state level, each state has a constitution which can be as different from one to another as it pleases the residents, so long as it does not contradict the federal constitution (there's been some brouhaha over the state of Coahuila and Mexico City legalizing gay marriage, with opponents seeking to amend the constitution to force the issue). Each state has a governor and a unicameral legislative. The Supreme Court in Mexico is made up of 11 judges.
At the federal level, the system is similar to that of the US (not coincidentally, post revolutionary governments the world over love to cheat off of the US constitution). On the legislative side there's a bicameral legislature. During the government of the PRI, both were pretty much rubber stamps to presidential initiatives but have gained considerable power over the last two decades after the PRI majority in congress disappeared. Congress is the lower house, and is generally the "rowdy" bunch (with occasional—mind you, occasional—fisticuffs), a bunch of 300 elected congressmen and 200 "plurinominales" or "plural votes", these last are accorded to each party proportionally to their share of votes in the latest popular election. The party fills these seats as they like. Attempts to winnow this ungainly number have been... difficult. The Senate or upper house is much smaller at 128 and generally more serious (which isn't saying much; it's distressingly common for legislators in either house to seize the tribune).
After The Mexican Revolution, for much of Mexico's 20th century history, any real power rested with the President, a six year virtual dictator with supreme power over the PRI (the main political party from 1920-2000) and the government. Nowadays, the president still has a great deal of power... but it's the legislative who's boss. The thing about the six year dictators is that most of their power was never codified, and in fact it's the legislative that holds a lot of the weight. This is something the legislative is just figuring out in its adolescence, and the result looks rather like the US in the period between independence and the Civil War (you know, when brawls on the House floor broke out every now and again, and a Senator was beaten within an inch of his life, but the legislature controlled all).
Presidents, Governors and Senators are elected for six year terms. Every Governor and the President are barred from reelection once his/her term is up. And in the President's case, his office is strictly a one-term affair.
Mexico is a multi-party democracy, and since 1997 (when the PRI weakened) no party has held a majority in either house, so most legislation has to go through on a case-by-case basis. From the 1929 elections to the 2000 elections, all presidents were from the PRI. Parties are supposed to be funded by the taxpayer.
The four major parties are:
Although technically founded in 1946, PRI is the immediate soft reboot and direct heir (membership and whatever they called a "platform") of the PNR, a party first started by military men in 1929 to control the violent excesses left from The Mexican Revolution by forcing every important politician and leader in Mexico at the time to join and line-up or else. The organization has a deliberately ambiguous "big tent" philosophy and platform, accepting people from any political viewpoint as a means to broaden its appeal. For most of the 20th century, the PRI and its previous incarnations were the party in power through all of Mexico, winning every presidential election since the 1920's until 1994. To achieve this control the party was not above resorting to dirty tacticts to manipulate the results. However PRI would allow itself to lose occasional mayor or governor race to give the outward appearace of democracy.
After 1994 with the economic crash, the party was floundering without strong leadership and internal divisions making unified candidacies difficult, which led to the unprecedented and catastrophical (for the PRI) defeat against the PAN in the 2000 presidential elections. Unable to take a hint, the party kept up its outdated image and inner squabbling and in the presidential elections of 2006 suffered a terribly poor showing (the first time they came up third in such elections) because the Party president forced himself as candidate and earned the ire of all the other pre-candidates.
Finally straightening up their act, their fortunes improved: after the 2009 midterms, they gained a plurality in the lower house, made into a de facto majority via their alliance with the Ecologist Green Party (PVEM). This same alliance made it possible to eventually win the 2012 national elections and hence, held the presidency once more after twelve years out of office. Though explanations for this startling comeback are many, it's mainly attributed to the 2007 electoral reform, anger at the world economic crisis, a lackluster performance by presidents from the PAN, and several PRI state governors using decades old and brand new vote buying techniques that the authorities can't or won't stop.
The PRI candidate elected president in the 2012 elections, Enrique Peña Nieto, went full tilt into deep economic, energy, communications, labor and educational reforms that many find too much too soon and even controversial. Others argue that Mexico has been in dire need of these changes for at least 25 years and that the speed of change is just an inevitable catching up to date. On the 2018 elections, the PRI suffered a monumental defeat when, despite ther alliance with the aforementioned PVEM plus the New Alliance Party (PANAL), they repeated their 2006 results and their candidate, former Peña Nieto cabinet minister José Antonio Meade (although himself not a member of PRI), came up third with 16% of the vote, in the worst result in the history of the PRI.
In 2000, the PAN made history and headlines around the world by winning the Mexican presidential elections, making it the first time in over a century that an opposition party candidate (Vicente Fox) installed himself in the Mexican presidency with due process (that is, not resorting to a coup). The fact that it was against an incumbent PRI famous for its formidable electoral machine and its resources embezzled from the state to fund its campaigns among other irregularities made the victory against a party that had won every presidential election since The Roaring '20s all the more astounding. Unfortunately a lack of experience, an unwillingness to exercise authority, and the colorful behaviour of Fox often stained this genuine triumph for democracy.
In 2006 PAN won again the presidential elections, but this time, its candidate (Felipe Calderón) won by a ridiculously narrow margin in the last minute after trailing in the polls the rest of the time, and Calderon's authoritarian and aloof style, along with a poor economic showing due in no small part to the 2008 world crisis and the ramping up of street violence and crime due to the start of the 2006 drug war in Mexico did very little to improve the party's image. All this led to its candidate (Josefina Vazquez, the first female presidential candidate fielded by any of the major parties, i.e. the first woman with any realistic chance of winning) being relegated to a third place in the 2012 presidential election which went to the PRI candidate. Their 2018 candidate, Ricardo Anaya, finished a distant second place.
Nowadays they're a hodgepodge of "tribes" ranging all over the left. This internal division has grown particularly nasty, since the two largest tribes basically accused each other of stealing the internal elections.
The 2006 runner-up presidential candidate, Andrés Manuel López Obrador had firm control over half the party, despite having alienated most of the middle class with his messianic, sometimes confrontational style, causing a reduction in voting preferences to 14% of the electorate. He was opposed by the more traditional "Chuchos", and together they divided the party with no clear reconciliation in sight. López Obrador was PRD's candidate for president again in 2012, but after also losing that election, he broke away for good from the PRD, forming with his followers yet another left-wing party called MORENA (see below). For the 2018 elections, the left-wing party did the unexpected move of allying with the right-wing PAN, whose candidate, as mentioned before, finished a distant second place.
Other parties - There's a sundry list of smaller parties, including the PVEM (Ecologist Green Party, which is the ecologist party of Mexico but is ironically promoting the legalization of death penalty, and has been criticized as being the front of a family mafia), PT (Labor Party), etc. None of these have yet gotten a governorship; however they have attained several seats in Congress and the Senate both at state and federal level. Normally they're too small to even act as kingmakers in congress, but these "stooge" parties practically exist only to keep the minimum share of popular votes necessary to keep their right to receive federal funding (i.e. pocket the money). So far only one of these parties has managed to actually gain a solid foothold in Mexican politics: the center-left Convergencia party, later restructured into Movimiento Ciudadano, thanks in great part to MC-aligned Enrique Alfaro having done a particularly good job at ruling the municipality of Tlajomulco, Jalisco, which just happens to be part of the Greater Guadalajara area, which in turn happens to be the second largest metropolitan zone in the country, thus allowing the MC to gain an actually solid following in the state of Jalisco — solid enough to almost win the 2012 state governor elections. 2015 also saw an increase on independent candidatures, culminating in the victory of Pedro Kumamoto as state congressman in Jalisco, and the victory of Jaime Rodríguez "El Bronco" Calderón as governor of the state of Nuevo León (home of Monterrey, an industrial boom city almost the same size as Guadalajara).
Currently the most pressing issue nation wide is insecurity and the violence brought in by the infighting between drug cartels and federal government efforts to curb these cartels. For decades the police forces have been neglected at local and state level resulting in police forces that are substandard even by Third World expectations. Municipal and state police are underpaid and poorly armed, being no match for the drug cartels who supply themselves via illegal gun smuggling mostly through the United States. These leaves the federal government forces (including the Army) virtually alone to fight the huge mafias that developed over the country for decades sometimes protected by local state governments and bureaucracies.
Traditionally, the biggest issue has been extreme poverty and corruption. Income-wise, Mexico is among the most unequal nations in the world and decades of government intervention to supposedly help solve it have only resulted in corruption. The country has spectacularly diversified and modernized its economy in the last two decades and is an industrial powerhouse. It is the nation with the most free trade agreements in the world. However, due to lack of reforms in its ridiculously outdated set of bizantine regulations, Mexico's economy rarely grows beyond 3% a year.
Energy has always been the most sensitive (if most of the time dormant) issue; in Mexico oil production, from exploration, to refineries is by law exclusive state property and seen by many people as a bona fide national symbol not to be let into private hands, let alone foreign. This has stalled much needed reform in the energy sector and squandered any economic advantage Mexico might enjoy as an oil producer. Oil production is expected to last around 20 years more (as of 2012) before it exhausts.
The electoral reform of 2007-2008 (made to accomodate the PRD due to that party's inconformity with the 2006 presidential election results) made the process more expensive and alienated voters and the media monopolies by flooding the airwaves with advertisements the government forced the media to show.
After President Peña Nieto's reforms, investment in energy is expected to increase in the mid term future, however it's still anyone's guess if it will affect prices of energy in the country. It remains to be seen if the reforms create any worthwhile improvements in the areas of the economy and government that don't cater to the syndicates/oligpolies. López Obrador rise to the presidency may be another spanner in the works for these plans as he has repeatedly stated he wants to return all oil production firmly into state hands.
The Zapatistas (EZLN) are an issue that shrinks and grows day to day. Taking their name from revolutionary leader Emiliano Zapata, the Zapatistas are a group of self-defined indigenous people in the deep southern state of Chiapas who rebelled prior to 1994 in order to gain international recognition for the plight of indigenous peoples. Though not a political party, the Zapatistas have been accorded governmental autonomy for the municipalities they control, making their tribal council rule a political force at least locally. The cliff notes version of their plight is this: for much of it's pre- and modern history, Mexico has never known what to do with "the indigenous problem": remote indigenous populations with high levels of poverty, illiteracy and no infrastructure, leading to systematic neglect and abuse. Following the passage of NAFTA, and specifically the adoption of a provision which would void all claims by indigenous tribes to their ancetsral lands, they rebelled (with actual arms and a few battles against anti-Zapatista paramilitary groups and the army). However, after the initial gains made negotiating with the government, they dropped off the radar when the negotiations fell apart and have mostly focused on internal matters, barring the occasional "Zapatista tour" or cryptic pronouncement by their leader Subcommander Marcos. There are rumors of an entrenched, corrupt political elite among the ruled municipalities, while ironically, the government's pumping money into improving the situation for all non-Zapatista municipalities has given them a marked improvement in standard of living, though this might only be to show up the Zapatistas. Despite this, the EZLN still enjoys a great deal of popular support in their municipal communities, who prefer the independence they to the greater resources of the federal government.