A developer may specialize in a certain video game console (such as Nintendo's Nintendo Switch, Microsoft's Xbox One, Sony's PlayStation 4), or may develop for a number of systems (including personal computers and mobile devices). Video-game developers specialize in certain types of games (such as role-playing video games or first-person shooters). Some focus on porting games from one system to another, or translating games from one language to another. Less commonly, some do software-development work in addition to games.
Most video game publishers maintain development studios (such as Electronic Arts's EA Canada, Square Enix's studios, Activision's Radical Entertainment, Nintendo EAD and Sony's Polyphony Digital and Naughty Dog). However, since publishing is still their primary activity they are generally described as "publishers" rather than "developers". Developers may be private as well (such as how Bungie was, the company which developed the Halo series exclusive to Microsoft's Xbox).
In the video game industry, a first-party developer is part of a company which manufactures a video game console and develops exclusively for it. First-party developers may use the name of the company itself (such as Nintendo), have a specific division name (such as Sony's Polyphony Digital) or have been an independent studio before being acquired by the console manufacturer (such as Rare or Naughty Dog).
Second-party developer is a colloquial term often used by gaming enthusiasts and media to describe game studios who take development contracts from platform holders and produce games exclusive to that platform. These studios may have exclusive publishing agreements (or other business relationships) with the platform holder, but maintain independence so upon completion or termination of their contracts are able to continue developing games. Examples are Insomniac Games (originally a 2nd party for Sony), Bungie (originally a 2nd party for Microsoft) and Rareware (originally a 2nd party for Nintendo).