The game of Botticelli is played like this:
One player, the chooser, thinks of a famous person (living or dead, real or fictional). The chooser says the initial letter of that person's last name (or of the person's only name if there's only one). The other players then try to guess who the chooser is thinking of, in a roundabout fashion: a player thinks of a famous person whose last name begins with the given letter, and gives a clue about that person. If the chooser can guess who the player is thinking of from the clue, that turn is over and another player can give a clue for another famous person with the appropriate initial. If the chooser can't guess who the player is thinking of, the player gets to ask a yes/no question about the original famous person chosen.
As usual, an example should clarify things:
Andrea [thinking of Copernicus]: I'm thinking of a famous C.
Bernard: Did you discover America?
Andrea: No, I'm not Christopher Columbus. It's okay to give harder clues.
Colette: Are you merry in the nursery?
Andrea: No, I'm not Santa Claus.
Colette: That's not who I was thinking of!
Andrea: It fits the clue, though. You'll have to come up with another clue for your person.
Colette: Hmph. I think Santa is more jolly than merry... Okay, did you smoke while someone fiddled? —And don't say Nero, that doesn't start with C.
Andrea: I give up, who are you thinking of?
Colette: Old King Cole. Here's my yes/no question: Are you fictional?
Bernard: Did I float and sting?
Andrea: Oh, damn—I know you're talking about Muhammed Ali but I don't remember his real name.
Bernard: Cassius Clay. Are you alive?
[Some time later...]
Colette: Did you go mobile for art?
Andrea: I dunno.
Colette: Alexander Calder. ...I get a yes/no question but I forget what we know so far. Can you review it?
Andrea: Not fictional, not alive, male, not an artist, European, pre-19th-century.
Colette: Were you a scientist?
Bernard: Did you know the sun didn't revolve around you?
Andrea [assumes he's got it, but maybe he has someone else in mind]: Who are you thinking of?
Bernard: Nicolas Copernicus!
Of course, players don't have to alternate coming up with people—a player can take multiple turns in a row if other players are stuck.
In the strict version of Botticelli, the player's guesses have to be people who fit all of the known yes/no criteria so far. That is, in the example above, after the first yes/no question, no player could give a clue for a fictional character. It's possible to play this way, but very difficult, especially after four or five yes/no questions have been asked; I don't recommend the strict version unless you're really good at coming up with people who match a set of categories.
The yes/no questions in the example are fairly standard ones. Note that a fictional character is considered "alive" if ta is alive at the end of the fictional work in which ta appears.
Thanks to seclar for reminding me to do a column on this game...