Blackjack hands are scored by their point total. The hand with the highest total below 22 wins, going over 21 is called bust. Cards 2 through 10 are worth their face value, and face cards (jack, queen, king) are also worth 10. An ace counts as 11 or as 1.
The goal of each player is to beat the dealer, by having the higher, unbusted hand. Note that if the player busts he loses, even if the dealer also busts. If the player's and the dealer's hands have the same point value, this is known as a "push", and neither player nor dealer wins the hand.
After bets are placed, the dealer deals the cards. The dealer gives two cards to each player, including himself. One of the dealer's two cards is face-up so all the players can see it, and the other is face down. A two-card hand of 21 (an ace plus a ten-value card) is called a "blackjack" and is an automatic winner. If the dealer gets Blackjack then all the players lose.
Each player takes it in turn to stick or twist (get another card). The target being 21, or as close as you can get to it. Once all the players have completed their turns it's the dealers turn. Most casinos will follow the rule that the Dealer will twist until they reach 17 (or above) then they must stick (If they are not already bust).
Depending on the quality of the winning hands (Blackjack being the best) different returns may be paid by the casino, but this varies from evens (1:1) to higher returns for Blackjack.