Blackjack is the most popular table game in the casino which offers excellent odds. The house edge is a tiny 0.5%. That's just about the best odds on any game as you're likely to find, making blackjack very attractive for those willing to put in just a little bit of work to learn the strategy (a couple of hours or so).
Surprisingly, most players won't make this effort. I've played hundreds of hours of blackjack, and have rarely seen anyone who knew proper basic strategy. By guessing at the proper strategy, they're willingly giving the house a 2% advantage -- meaning they lose four times as much. It's throwing money away. Don't be one of those people.
Play begins with the right-most player ("1st base") and continues player by player to the left. Your objective is to beat the dealer's hand; the higher hand wins, as long as it doesn't go over 21 (bust). If both of you bust, you still lose. This is why the casino has the advantage in blackjack.
When it's your turn, you have the following choices:
Once you've played your hand, that's it; play will not come back to you. Each player gets only one turn per hand. You can hit as many times as you want, but once you're done hitting, that's it.
After each player has played, the dealer plays her own hand. She flips the hole card over first so everyone can see both her cards. The dealer must hit (take cards) until she has 17 or higher. That's the rules; the dealer isn't allowed to make decisions on whether to hit or stand depending on what the players' cards are. If the dealer could vary her play depending on what the players have, the house edge would be so high that no one would play. Note that on most tables, the dealer will hit her 17 if it's a soft 17, meaning that it has an ace that counts as 11, and is therefore unbustable.