I know of at least three entirely different games called The Name Game.
One of them is used for introductions and theatre warmups: everyone stands in a circle, and one person says their name paired with an adjective that starts with the same letter—"Frantic Fran," for instance. As they say this, they make a gesture or motion that suggests the adjective. (Such as waving their hands frantically over their head.) The rest of the group joins them in repeating the phrase and the motion three times. Then the next person comes up with an adjective and motion for their name, and so on. For each new person, you repeat all the people who've gone before. By the time you get around the circle, you know the names of the first people pretty well.
The second Name Game involves putting names of famous people on headbands or taping them to people's backs. It's to be played at a party; you try to figure out whose name is taped to your back (or is on your headband) by asking other people questions. I dislike this game, largely because I have bad associations with it; I once went to a Halloween party wearing a rather good Invisible Man costume (head entirely bandaged, plus trenchcoat and gloves and hat), and they were playing this Name Game, and some wag figured out who I really was and put my name on a card taped to my back, thereby ruining the disguise.
The third Name Game is the best of the lot, and is generally the one I'm referring to when I mention "the Name Game." A similar game has been marketed under the name Celebrity Taboo, but as with most attempts to box and market a parlor game, Taboo isn't as much fun as the original.
The Name Game begins with everyone writing the names of famous people on slips of paper. Any famous person (living or dead, real or fictional) is allowed, but as with Botticelli, the people named should ideally be people everyone in the group has heard of but wouldn't necessarily think of immediately. It's not much fun to try to get someone to say "Queen Boadicea" if the other person doesn't know the name—or, for that matter, if you don't know the name. (Resist the impulse to use the names of people in the room, unless they happen to be famous.) Each player should submit about eight names. Fold each slip of paper in half to hide the name on it, and mix all the slips together in a hat or other container.
Now the game proper begins. One player pulls a name out of the hat and gives clues to another player to get the other player to say the name on the slip. When the clue-receiver says the correct name, the clue-giver discards the slip and picks another. The process repeats until the time limit is up, at which point it becomes another clue-giver's turn. Each turn lasts thirty seconds; someone should act as timekeeper each turn, using a stopwatch, a sand timer, or a watch with a second hand. (Some versions of the game allow a full minute.)
There are rules about what kinds of clues are allowed. The standard rules at Swarthmore say that a clue cannot contain the name of any person, nor can they contain direct "rhymes-with" clues. (For instance, cluing "Philip Glass" you could say "His last name rhymes with the green stuff that grows on lawns," but you couldn't say "His last name rhymes with 'grass.'") Direct spelling clues are also illegal (you can't say "His last name is spelled G-L-A-S-S" or "His last name begins with G"). Book titles are allowed as long as they don't contain the names of people, so "He wrote Twelfth Night" is valid, but not "He wrote Henry V." Names of musical groups that don't contain people's names are generally allowed in clues; but the names on the slips of paper should be the names of individuals and not of groups.
There are variations on the allowable-clue rules. Some people play that place names aren't allowed if the place in question is named after a person (so "He lives in Washington, D.C." would be illegal). I feel that rule is too hard to enforce; some places (like Nottingham) are not widely known to be named after people. Instead, I would say that places named after people are acceptable as long as the person in question isn't directly relevant to the clue (so you can't say "Washington, D.C. is named after him"). In my favorite (very strict) version of the game, clues can't contain any proper nouns at all. (Direct rhymes-with and spelling clues are still illegal, of course.)
At the end of their turn, both the clue-giver and the clue-receiver get one point for each name correctly guessed within the time limit.
Order of play is a little complicated. The first time around the circle, each person gives clues in turn to the person on their left. The next time around, each person gives clues to the person two seats to their left. The next time around, three seats to the left, and so on. You can either play 'til you run out of names in the hat, or until you've exhausted all possible combinations of clue-giver and clue-receiver.
Once you finish going through all the names in the hat, the fun really begins. You put all the names back into the hat and start round two, in which you use the same set of names, but clues must be no more than three words long.
In round three, you don't get to use words (or other sounds) at all, only gestures. Standard Charades rules apply.