"Originally independent, since the release of
Pokémon Red and Green in 1996, Game Freak has been a
second-party developer for
Nintendo, developing games exclusively for their hardware while remaining a separate company. The company currently employs a staff of 71 as of April 2011"
http://bulbapedia.bulbagarden.net/wiki/Game_Freak
Second party developer is a colloquial term used by gaming enthusiasts and media often used to describe two different types of game development studios:
- Independently-owned studios who take development contracts from the platform holders and what they produce will usually be exclusive to that platform.
- Studios who are partially or wholly owned by the platform holder (also known as a subsidiary) and what they produce will usually be exclusive to that platform.
GameFreak is the latter, minus the occasional iOS thing.
Edit: And to round it off, SEGA and Capcom are third party, though it's almost easier to show stuff for them because they gave all permissions to Sakurai already. With 2nd Party, it's an ongoing "Hey, can I do this?"