I always wonder if the whole 1st party and 3rd party games were like 1st and 3rd world countries.
What would be 2nd party games? Homebrew? Unofficial dark dev group? Soviet Russia gulag, potato, and roulette to kasplode video game?
AFAIK
1st party games - Games developed mostly by studios that are owned by the console maker companies. Like the Sonic 1, 2, 3 games that were made by either Sonic Team or STI, studios created internally at Sega branches. Or games that will be released by the various studios that Microsoft have acquired in the last years and will be exclusive to Xbox and PC. Some games may have lots of assets outsourced to external companies, but if they had involvement from internal staff or close supervision they are labeled as 1st party games.
2nd party games - Games made mostly by studios not owned by the console manufacturer, but these games were made by contract/partnership between the console company and the softhouse that made the game. These are the case of Mario Tennis games, they are made by Camelot, published and owned by Nintendo, but Camelot themselves aren't a Nintendo owned company (funny enough their original parent company was SEGA).
3rd party games - Your standard game publishing deal, a given softhouse may self publish or work under a publisher to create and release games. The publisher have to pay fees to console companies to have the license to launch the game to a given system. In this publishing model the console companies receive also a small fee on every copy of the game sold to their system.