Hinopio got it absolutely right.
A third party will make games for all systems. Like Namco (Tekken for PSX, Ridge Racer for N64, Soul Calibur for DC etc).
A second party is usually kept exclusive to one company, but could always go and make games for other systems. Rare is a second party to Nintendo (they had a few games on the Sega Megadrive though aparently), Square is a Second party to Sony now (but previously for Nintendo), and Sonic Team and Bizarre Creations are Second partys to Sega.
Namco used to be almost second party to Sony, but now they've gone completely 3rd party.
And a first party is the company themselves, like Sega, or Nintendo. Sony doesn't have any first party development I think. Just very, very close 2nd partys.
Hope this sheds some light on the situation.
This victory strenthens the soul of Jonaskin!
-Soul Calibur, Sega Dreamcast.
Greatest fighting game ever.