In blackjack card counting is keeping track of the number of different sorts of cards that have come up. When a deck has a lot of tens, jacks, queens, kings and aces in it it's advantageous to the player. When it has a lot of low cards, 2's through 6's, it's more advantageous to the house than usual.
A card counter will keep track of these numbers usually with one running counter. If they see a 2 through 6, or a red 7, they will add 1 to a number in their head. If they see a ten through ace they'll subtract a number. As the count gets high (depending on the system, somewhere above 12), they will increase their bet significantly to take advantage of the edge they have.
I wouldn't suggest getting into card counting. Firstly you'll need to have perfect basic strategy down, which takes some time to learn. You'll have to know it like the back of your hand too, so you are able to keep count and pay attention to other things while playing. You'll also need to learn some deviations from basic strategy based on your count. You will occasionally make different decisions than you otherwise would since you know more about what's likely to happen than most players.
Then you'll need to learn how much to bet. Since variance is very high in blackjack your biggest enemy in playing is gamble. At your best you've got a small edge, and with high variance it takes a long time for that edge to play out into something concrete. If you don't have a sufficient bankroll the house will simply bust you before you can play enough hands.
The only successful blackjack players I've known have had to play ridiculously high limits to win. To play that high though, you'll need to risk a few hundred thousand dollars. Even then there's no guaranteed win. You have an edge, but it could very easily fall apart on you. Variance is vicious.
The blackjack pit is filled with wanna-be card counters and the casino loves it. Very few counters get good enough to be beating the game, but the possibility of the game being beaten fills the seats with a ton of players that otherwise would never play a table game.
It's also important to realize that once you get your game together and you are a winning player and you're just ready to cash in on your investment of time and energy, the casino will simply back you off.
If you're smart enough to win at blackjack you're smart enough to realize that it's a grind. I've never met a pro BJ player (and I know a few) who suggest taking it up.
It doesn't seem like it is any different in method than card couting for poker...
You don't count cards in poker. The deck is shuffled after every hand so there's nothing to keep track of from hand to hand.