Giovanni first appeared in Pokemon Red and Blue (Red and Green in Japan) as the Final Gym Leader. (Read what a gym leader is here.) When playing the game, one would reach Viridian City right after leaving their home town. The gym is the largest building but at the beginning of the game, one would find that the gym is locked! Playing through the game, Giovanni would be met multiple times, but most notably at Silph Co. where he would be revealed as the Boss of Team Rocket. This malevolent figure would be seen again, as the final Gym Leader! Imagine what a shock this would have been to the individual playing this game for the first time. How could the leader of a sinister organization be the leader of a Pokemon Gym? (This is discussed in depth in the Gym Leader section.)
Aside from later versions such as Yellow and FireRed/LeafGreen, Giovanni does not make this same appearance. He is able to be fought in the newest game, HeartGold and SoulSilver, only through a special event. He does, however, some brief appearances in other games including Pokemon Stadium and Pokemon Puzzle League. In Puzzle League, you are actually able to chose him as a playable character. This game is unlike the other Pokemon games because it is similar to that of tetris and is a puzzle-fighting game instead.
Giovanni appears in games only when there is a need for the gym leaders or a criminal mastermind. He is not a beloved character who returns every version, and is predominately excluded from the newest games while the eigth gym leader is now Blue, the rival from the first game. (Commonly referred to as Gary.)