A subtrope of It's Raining Men, this is a favored tactic of the Space Marine. You need to get your troops to the battlefield. You also need to inflict damage to the enemy. Why not do both at the same time?
A Drop Pod is usually launched from a ship—usually in space—like a bullet after being loaded with troops, preferably in Powered Armor. Sometimes it has its own thrusters, allowing it to adjust its course in flight. When it hits, it usually performs a Ground-Shattering Landing, softening up the enemy enough to allow the troops to disembark safely.
Note that while a Drop Pod is nearly always used to damage the enemy, it doesn't always. The Starship Troopers version typically disintegrated, creating chaff to protect the trooper from anti-air fire, by the time it reached the ground.
Compare Drop Ship, which can fly back up to the orbiting ship afterward. If launched against another ship instead, you're looking at a Boarding Pod. If you're expecting everyone to die, you might be considering a Colony Drop instead.
Like many Space Marine tropes, the Trope Maker here is Starship Troopers. Little has changed since Robert A. Heinlein invented the trope. It is worth noting that in the book, the pods were launched by being fired out of a tube like torpedos, in the oppoosite direction the ship was orbiting, to help them slow down enough for reentry.
In the New Jedi Order novel Rebel Dream, special guest stars Wraith Squadron unveil their latest in Awesome yet Practical devices: one-man covert drop pods (really little more than shells) that are designed to mimic spaceborne debris falling to a planet. They proceed to use them to insert an infiltration team onto recently-captured Coruscant despite the massive enemy fleet presence.
Atari's Middle Earth pinball has futuristic explorers use bulb-shaped chrome pods to land on a Lost World filled with dinosaurs and monsters.
The Imperium's Space Marines use this as their primary tactic, and can even drop Dreadnoughts (twelve-foot-tall battle sarcophagi). Many other races have similar abilities. The tactic of unexpectedly inserting your troops behind enemy lines is known as "deep striking," whether it is done with Drop Pods, teleporters, or (in the Tyranids' case) wombs fired from space.
The Space Marines themselves do not rely on the Ground-Shattering Landing to clear the way. Instead they launch two waves of pods, the first wave contains pods equipped with automated guns to clear the landing area for the second wave (which contains the Marines).
Da Orks, being who they are, kick this one up a couple notches in the form of Roks. They take an asteroid, add engines, guns, and armor to it, fill it with Orks, and ram it into a planet. The Rok either causes a huge ruckus on impact and unleashes a horde of greenskins, or simply explodes and kills everything in the vicinity. Orks don't care which.
Champions supplement Gadgets. One of the title items was the "Jump Shell", said to be used by the U.S. Marine Corps to insert troops into the battlefield from orbit. Less advanced interim models would be dropped from supersonic transport planes.
Classic Traveller Adventure 1 The Kinunir. Some of the Marine troops on board the title ship are trained to use jump capsules to descend from orbit and land on a planetary surface.
BattleTech has Mech-sized drop pods, as well as power armor sized ones. They are usually dropped directly onto a possible landing zone to secure it, so that the DropShips can arrive to bring in the real army.
Halo gives us the HEV, or Human Entry Vehicle, designed to shoot one soldier at the battlefield. These are launched in large numbers; the idea is that if one gets shot down, you lose one troop rather than the dozens that would be aboard a dropship. The troops who use this are the ODST, also called Helljumpers, and are the second best the human military has to offer, after the Spartans. In Halo 2 the In Amber Clad deploys the Chief and a squad of Helljumpers onto Halo Installation 05 this way, prompting this remark from Cortana:
The Tiberium saga of Command & Conquer features this in two iterations of the Support Power variety. The Tiberian Sun iteration crash drops a handful of veteran low-tier GDI infantrymen on the battlefield. The Tiberium Wars variant, on the other hand, lands a commander a few squads of veteran GDI Zone Troopers which are powerful enough to turn a pitched battle into a devastating curb-stomp.
On the other hand, the Tiberian Sun pods have a function that clears the landing area...with gunfire. Helpful if you're dropping your boys into a hot zone and need to take out a cluster of high-level enemies. The Tiberium Wars pods land gently, slowly, and are generally sitting ducks until the Zone Troopers get out.
In Star Wars Battlefront: Elite Squadron, one may utilize the escape pods on capital ships as a quick way of getting to the planet's surface to attack.
In Mega Man Legends 2, a drop pod has to be built in order for MegaMan to reach the surface of Forbidden Island due to the violent blizzard.
There were drop pods in Battlefield 2142 which could be launched from air transports, Titans, APCs and the player would spawn from one if you selected a spawn beacon for your spawn point. Before the most recent patches, they could be used to destroy vehicles, and a glitch allowed you to fly around the map.
In the campaign of Starcraft II the Terrans use drop pods, the mercenary compound brings merc units to the field in seconds using them and a tech upgrade allows the barracks to do the same. In addition the Zerg have "Sacs" that fill the same function in certain missions.
In the Quake II and Quake IV universe, the drop pods are almost identical as how they are in Starship Troopers. In II, the player starts when he misses his drop zone and ends up and the far side of enemy territory. In Quake IV, the player uses a drop pod to make a pin-point attack on a Strogg command centre. Coincidentally, both landings are crash landings due to malfunctions in the pods' systems after they're clipped by another podnote which, in II's case, saves Bitterman's life, most of the other pods being shot down by the Big Gun's EMP.
PlanetSide has the HART drop pod. The HART shuttle flies above the planet, and ejects the soldiers out of the bottom in large metal pods with small thrusters on them. Once the pod lands, it unfolds and releases the soldier. Planetside 2 drops the HART, but allows players to use the drop pods by using the Instant Action utility, through squad spawn beacons, or spawning on a squad leader. The PS2 drop pods are player-controlled, and have some maneuvering thrusters - most players use them to crush aircraft or land on the heads of infantry
Used in the FEAR series to drop down armour onto the battle field.
In Awesomenauts, this is how your character enters the battle at the start of a game, or when respawning.
Cortex Command has drop crates which are essentially the same thing. Said crates only have 2 exits and can end up in situations where both are blocked, resulting it the crate destroying itself so things inside it can escape.
In the first Dawn of War and its expansions, any Space Marine unit could be redeployed via Drop Pod, though it required a clear landing area. In Dawn Of War II, this is how your squads enter the map, and they can summon another one that instantly replenishes your troops (and this time, can be brought down on enemies' heads).
During the Halloween invasion of Superhero School Whateley Academy in the Whateley Universe, ARC provided support by sending a couple dozen fighters in power armor using drop pods from a space station overhead.
Roughnecks: Starship Troopers Chronicles had the troopers first embark onto Drop Ships, and then they would be dropped from there in pods, presumably to allow the mothership to stay clear of Bugdefensive fire. One episode has Rico get cast adrift in one of these due to a glancing blow in mid-drop. He is eventually saved by Carmen, but not before a Clip Show can ensue as Rico's life takes nearly 22 minutes to flash before his eyes.
The mundane version doesn't have a Ground-Shattering Landing, but is effective; during World War II military gliders were used to drop troops, jeeps, light artillery and even light tanks, achieving a superior concentration of forces with better support assets than a parachute drop.