Giovanni is evil. Okay? You know, the eternal evil dude that you need in a cartoon. (There's always one). And... well... that's what he does. Yeah.
He's the leader of Team Rocket and the Viridian City gym leader (he uses ground Pokèmon), and in red/blue/yellow you'll have to beat him two times, one in the Team Rocket Headquarters in Saffron and one in his gym. In Gold/Silver (which takes place three years in the future) Team Rocket is still there, but now under those "rocket executives" who claims that he'll be back to let Team Rocket rise again. He never appears, though, and his gym is run by Blue (Gary).
In the show Giovanni doesn't appear all that often, and he isn't referred to as Giovanni, but as "the boss" by his employes. Until the episode "Battle of the badge" we never see his face either, it is always hidden in shadows and he talks in a voice changer. The mystery about the man in the red tux is solved, however, when Gary challenges him. You know the deal, he beats Gary with his "mysterious pokèmon" (Mewtwo, and I'll get back to him later anyway) and runs away with it and leaves the gym in the hands of the exellent battlers Jesse and James (let's just say that one of them thundershocked... um... some stone pokèmon in "The ultimate test), who of course looses to little Mr. "I'm gonna be a pokèmon master", and the gym is destroyed in the end of the episode.
Usually he appears to yell at Jesse and James, but he also were in the episodes with Butch and Cassidy (well, at least two of them). It's clear that he favors the last team; he sets them to run scheme and decides not to tell Jesse and James because they'd botch it, and he bails the other two out of jail themselfs (at least according to Cassidy).
He also appeared in the 1st movie, where his scientists created Mewtwo from the DNA of Mew. He told Mewtwo they would be partners and trained him and helped him become stronger, but at last he let it slip that he only wanted to control the pokèmon, and Mewtwo destroyed his headquarters somewhere in the forest and escaped to destroy all of humankind and all capture pokèmon.
He is also rumored to be the father of our dearest protagonist, but if this is right or not nobody knows, but it IS a fact that we don't know who Ash's father is, and that Giovanni did know Delia when he was younger. And when Delia is blond, his father have to be black-haired.
Something else worth to mention?
Well, he didn't start running Team Rocket, his mother, Madam Boss, did. She didn't love him all that much (of course), and... well, he's the one in charge now. He's probably somewhere between 35 and 45 years old, and has a passion for Persians (well, do YOU think Meowth will become Top Cat again anytime soon?)
And... well, some name fun (my Norwegian teacher told me): Giovanni is Italian (duh), but here are some other facts:
Giovanni = John = Johannes (Norwegian) = Ivan (Russian).
Then there is the meaning of his name, but I don't know and I won't steal it from other sites, so just... surf.