Blackjack is sometimes referred to simply as 21, as the aim of the game is to have a hand value that is closer to 21 than that of the dealer, without going over. Notice that you are only playing against the hand of the dealer, NOT against any of the other players at your table. The dealer has certain rules of play that they must abide by.
The basic idea of getting close to 21, or reaching it, and beating the dealer, is the same worldwide. However, from there on, there are a number of variations, especially between UK and US play, as shown in the table below.