It’s true that a true card counter probably shouldn't take a lot of notes or write things down.  But if you’re just starting out, I wouldn’t worry about it too much.  If you’re under ten years old, it’s harder to get away with that sort of thing.
assistant
People try all the time to spot card counters, but often it’s very hard to detect because all the players do a lot of things at once, like drinking, chatting, etc.  But most professional card counters are good at remembering the cards.  They count cards in their head, and use their memory and experience to remember the ratios of the remaining cards.  This is a skill that develops with practice.
assistant
I want to get better at playing blackjack.  What does it mean to be a card counter?
user
It means that you learn how to calculate the odds of the game, so that you’ll be able to get a larger amount of money from the casino without having to wager as much.
assistant
How do I actually count cards?
user
There’s a specific “technique” you should learn.  Some experts are better at it than others, and it helps a lot to be a “high-value” player (meaning you bet a lot on each hand).  You’ll also want to learn a lot about the mathematical odds, the relative probabilities of different events occurring.  For instance, if you hold 12 cards, it’s very hard to have a natural 21 (that is, a hand with cards adding up to 21).  But if you have a hand with 17, you can combine it with another hand to get a 21.  You might also like to know the odds of people making mistakes.  And when the deck is reshuffled, it’s important to know the odds of the dealer turning up a black card.  A good card counter would use this information to calculate the odds of different strategies of betting or folding, and would bet more when the odds were in his favor, and bet less when the odds were against him.
assistant
Don't I have to count how many of each card have already been dealt so I know how many are left in the deck?
user
A true card counter would never rely solely on memory.  It’s important to write down, on the table, the number of each type of card that’s been dealt out (e.g., 5 spades, 3 hearts, 7 clubs, etc.).  Sometimes these numbers will match what you expect, and that’s when you can use memory, but it’s important to do it in this way.
assistant
Won't I get caught if I'm writing everything down?
user