Giovanni is the leader of Team Rocket, the antagonists of the original games who tried to conquer the world with pokemon. Since their plans were unrealistic, Giovanni was also a part-time Gym Leader for Viridian City. He's not too difficult to defeat, since he mainly uses ground-type pokemon. I was a bit sad to never seen him comeback after the original game.
I liked his role in the anime better. You really felt his weight as this behind-the-scenes leader of Team Rocket. Jessie and James were lazy and incompetent, but Giovanni didn't mess around.
I also remember him in the anime movie where he attempts to control Mewtwo. Loved it!
Never watched the anime long enough to see his more expanded role.
His Japanese name is サカキ (Sakaki) which is derived from 榊 (Sakaki) a sacred tree in the Shinto religion that resembles the Earth Badge from his gym.