Corneria is one of the few planets, and most recognizable, that is home to majority of characters and populace in the Lylat System. Very much similar to Earth, Corneria is often a target for Andross' schemes as well as any other threats that try to cause terror, such as the Aparoids or the Anglars.
Corneria is the first level of the original SNES Star Fox and Star Fox 64. There are differences between the games, but they both serve the purpose as getting the player started on the game.
In the original Star Fox, before you began the first level, you chose one of three paths, all of which lead to Venom. Depending on which path you chose, the level layout for Corneria would change. The paths acted as a difficulty select screen, with one being easy, one normal, and one hard.
In Star Fox 64, the path that you took to Venom was determined by a set of requirements you complete during the first mission. After reaching the checkpoint, Falco Lombardi comes under fire by enemy forces, and you have to save him. If he goes down, you won't be able to access the second end boss and access to the "hard" route. Once you save Falco, you'll come across archways protruding from the water, and you fly through them. Falco will be impressed, and after you clear them, he leads you to the right through a water fall, going after the second boss. Defeating him earns you a "Mission Accomplished" and access to Sector Y (the start of the hard route).
If you failed to meet any of the requirements, you'll simply fight the regular boss in All Range Mode, and will only be allowed to go to Meteo.
Corneria came into play during Star Fox Assault when the Aparoids launch an invasion on the planet. Fox finds himself teaming up with bitter rival Wolf O'Donnel to help take down the Aparoid forces and General Pepper's assimilated ship. The planet once again served as the first level of Star Fox Command when the Anglars attack. This is where Fox reunites with Slippy Toad, and after defeating the enemy forces, leave the planet with a task in mind, based on which one the player chooses.
While Corneria was not the first level in Star Fox Assault, Slippy finds himself getting shot by an enemy and you have to save him not to long after you start playing. A funny situation, because it was the exact same deal in Star Fox 64, with Slippy getting your attention to help him. This was no doubt the start (probably earlier maybe) of player's hate for the character, which doesn't need to be explained at all, really.
In Star Fox 64, Corneria was the first level that players heard Peppy's infamous line/internet meme.
"Do a barrel roll!"