First party describes a game or studio owned by, or publishing for, the owner of the platform it appears on, such as Nintendo publishing a Nintendo developed game for its own console. Third party describes independent publishers and studios working across platforms.
The clearest case is a studio the platform owner owns outright. Nintendo internal teams, PlayStation Studios teams owned by Sony, and Xbox Game Studios teams owned by Microsoft are first party developers, and their games are first party games. The platform owner funds them, publishes them, and normally keeps them exclusive to its own hardware and services.
Between the poles sits the informal label second party: an independent studio not owned by the platform owner that makes a game published by that owner, often exclusively. The term is widely used by players and press but is not a formal legal category, and the same game is often described as first party by its publisher and second party by everyone else. What matters in practice is who published it and where it can be played.
Third party covers everyone else: publishers such as Electronic Arts, Ubisoft, and Capcom, and independent studios releasing on multiple platforms at once. Platform owners still certify and take a share of third party sales, but they do not own the studios or the games, which is what the party labels are really tracking.
In play"Mario games are first party, made and published by Nintendo for Nintendo hardware."