Fox: His father started the original Star Fox team (father was killed later). Rather brash, think Luke Skywalker in ANH in a fursuit and you've basically got it.
Falco: Mouthy, hot-shot bird who has left the team at least twice.
Krystal: love interest from beyond known space. May have been orphaned by Andross (big bad of series), cannot enjoy her sandwhich.
Peppy: only surviving member of the original team. serves as Obi-Wan Kenobi. Nearly sacrifices self in Assualt. tears were jerked. doesn't fight much anymore.
Slippy: the Odious Comic Relief of the team, should stay in a lab, since he's much better there.
Wolf: leader of rival team Star Wolf, foxes rival. Fox & Wolf have a begrudging respect for each other.
LeoN: cold, faintly sadistic assassin, rival to Falco
Panther: Cassanova-type who fell hard for Krystal.
Look up the Star Fox wiki.