In terms of games that involve strategy in a casino environment, blackjack is probably the one most well-known. Movies like 21 and books about teams of card counters taking on the casino has shot this game back into popular culture to a large degree. When you play online, card counting isn't a viable option, but there are still a lot of strategies that you can use to get a better payout rate than less-skilled individuals.
The way blackjack works is that the dealer is given one card face-down and one card face-up, and you're given two cards that you can see. The cards are all worth their own values with jacks, queens and kings always being worth 10. Aces can be worth one or 11, whichever you choose, and their value can change during a single hand however you want. You start off by basically having the option to either take another card (hitting) or keep your hand like it is (standing). Once your turn is over, it's the dealer's turn.
The dealer will then continue to hit until he or she has a total of at least 17. For either of you, going over a total of 21 will cause you to bust or automatically lose. However, if you bust and the dealer busts, then you still lose the hand, and this is the source of a lot of the house advantage. However, you get more strategic options with more complicated ways to play at your disposal than the dealer, and this allows you to make that house edge minimal (in many cases under 0.5 percent).
Aside from hitting and standing, you also get three other options that you can use at different points. One is splitting, and that allows you to "split" two cards of the same rank into two different hands, and you'll have to add an extra bet for the second hand. You can also double which means you'll hit exactly once without an option to do anything else afterwards, and this also means you double the size of your wager. The final "extra" option is surrendering, and this allows you to pull back half of your wager in exchange for losing the other half and having no chance to win the hand.
To learn to play this game well, you'll need to put in a lot of time learning what's called basic strategy. This is essentially a chart of information that will help you to know when to make different plays in different types of situations based on your own hand the card that the dealer has showing.
You can play blackjack for free at most online casino's but the real excitement starts once you place your first real money bet. Fortunately their are virtual tables to play for all levels of play, if you prefer to wager with low amounts you can take place at a low limit blackjack table or would you rather play high stakes at one of the high limit tables? Besides different wagering options you can also place bets on several blackjack tables at the same time with multihand blackjack.