After all bets have been placed, the dealer deals two cards to each player, moving around the table and dealing one card at a time. In a Nevada Deal game, the players' cards are dealt face down. The dealer then deals himself two cards -- one face up and one face down. In a London Deal game, the players' cards are dealt face up, and the dealer deals himself one card face up. In each case, the players are able to see one of the dealer's cards.
Each player in turn decides if his hand is close enough to 21 that it can beat the dealer's hand, based on the one dealer card he can see. If so, he stands, and the play moves to the next player. If the player wants to try to get closer to 21, he requests a hit, and the dealer deals him an additional card. If the player hasn't busted from the first hit, he gets to decide if he wants to be hit again or if he wants to stand. Play never moves to the next player until the current player either busts or stands.
Once all the players at the table have made their plays, the dealer reveals his second card (in Nevada Deal) or deals himself a second card (in London Deal) face up. If the dealer's total is 17 or higher, he has to stay. If his total is 16 or less, he has to hit. The dealer never gets to choose -- he must play by these rules for every hand. If the dealer busts, everyone at the table who didn't bust wins.
If the dealer doesn't bust, the dealer's hand is compared to each player's hand. The higher total wins, netting a 1-to-1 payoff for the player. A tie is a push -- the player gets his bet back, but wins nothing.
Getting a blackjack (an Ace and a card worth 10 as your opening two cards) is like an instant win. It pays at 3 to 2, or one and a half times the original bet. The dealer can also get a blackjack, in which case everyone at the table loses. If both the dealer and a player get blackjack, it's a push.