Blackjack game play is very simple. Many people think the object of the game is to get as close to 21 as you can without going over. But really you just have to beat the dealer. And to beat the dealer, all you need is a higher hand total than the dealer without going over 21. And if the dealer goes bust (goes over 21), as long as you don't go bust, then you automatically win.
Blackjack is a game of suspense. Wondering whether or not the dealer is going to bust is a high drama. I certainly think that it's a lot more dramatic than wondering whether or not you're going to hit a jackpot on a slot machine, although I know a number of folks who would disagree with me on that one.
Blackjack is also popular because it's a game of skill. Almost no other casino game offers players the opportunity to improve their odds by playing skillfully than blackjack does. Video poker sometimes comes close, but slots, baccarat, and roulette are all just completely random and have no skill element involved. (In fact, there are few bets in the casino worse than the roulette table, but that's the subject of another article.)
Blackjack is well-known as one of the only casino games that is "beatable". The legend of the MIT Blackjack Team that took Vegas for millions is well known all over the world now. Counting cards can be done by anyone with the discipline and patience to learn how to do it, although it's harder to do than you might think. I'm good at math, but I can't cound cards to save my life. And the real trick is being able to count cards and actually not get caught counting by the casino.
Blackjack is usually the best bet in the house, in terms of house advantage. If you're counting cards, you can actually have a slight advantage over the house. And even if you're only playing perfect basic strategy, you've reduced the casino's edge to about 0.5%, which are excellent odds compared to almost every other bet in the house.
Blackjack is the most popular casino game for all of the above reasons, but all of those reasons combined give blackjack a je na sais quoi that other casino games lack. "Je na sais quoi" is French for "a certain something" or "I don't know what". My friends who play blackjack regularly know exactly what this means in this context.
If you're interested in learning blackjack basic strategy, I recommend Blackjack Info.