A game made by nintendo or one of the companies nintendo owns is a first party game. A game made by another company for the wii is a third party game. Basically if it has nintendo characters like Mario, Samus, Kirby, Link (Zelda) then it is generally first party. Games that have no relation to any nintendo characters are usually third party
Super Mario Galaxy is made by nintendo and is first party.
The Conduit made by High Voltage Software and Published by Sega is third party.
People are usually concerned about the difference because nintendo makes the wii and makes software for their system but in order to have a larger library and more games for the system it is important for third parties to make games for the system as well. Nintendo systems have generally relied heavily on first party software and third parties have a hard time making games for the system because nintendo games are such high quality and hard to compete with and customers generally are more comfortable buying nintendo made games because they know they'll be good.
Sony and Microsoft's systems don't have that many first party games, they rely much more on third party companies to make games for their systems.