Blackjack like all other card games is constantly evolving, so there are quite a few variations of it out there. Here are some of them.
21st Century Blackjack - Player doesn't automatically lose, if he busts. If the dealer busts as well, it's a push (sometimes he has to also bust with a higher total).
Double Attack Blackjack - Blackjack only pays even money and the player has an option to increase his bet after seeing the dealers cards.
Double Exposure Blackjack - Both of the dealers cards are dealt out face up. Player loses, if it's a tie. Blackjack pays out 1:1 for even money.
Switch Blackjack - Player plays with two hands and can switch one card between the two hands. Blackjack pays out 1:1 and if a dealer gets 22 it's a push instead of a dealer bust.
Multiple Action Blackjack - The only real difference is that the dealer gets more than one hand, so the player pays multiple bets and plays his hand individually against each of the dealers hands.
Super Fun 21 - Player can split more often (up to 4 times). If a player has six cards with combined value of 20 he automatically wins the hand.
There are many more blackjack games out there that differ pretty much only by offering different bonuses, side bets and jackpots. You can find a few exotic blackjack games played online here.
You also have some special game rules that make the game different and the strategy behind it also changes with it. If you'd like to learn some strategy considerations behind different variations of blackjack, have a look here.