Mathematicians and blackjack experts have used computer simulations and statistical analyses to try to find a way to beat the house at blackjack. Through their studies, they have determined a basic blackjack strategy that gives the player the best odds against the house.
This basic blackjack strategy helps the player make decisions about when to hit and when to stand. By playing this basic blackjack strategy, the player can play on nearly even terms with the house. This basic blackjack strategy is the starting point for more-advanced card-counting strategies professional blackjack players use to gain an advantage on the house.

Basic BlackJack strategy

Below is the basic blackjack strategy for the typical rules found at online casinos. While you can't beat the house in the long term using this blackjack strategy alone, it does reduce the house's advantage considerably. You may want to print a copy of the blackjack strategy for reference.

Your Hand          Dealer's Up Card
             2  3  4  5  6  7  8  9  10 A
5,6,7,8      H  H  H  H  H  H  H  H  H  H
9            H  D  D  D  D  H  H  H  H  H
10           D  D  D  D  D  D  D  D  H  H
11           D  D  D  D  D  D  D  D  D  H
12           H  H  S  S  S  H  H  H  H  H
13,14        S  S  S  S  S  H  H  H  H  H
15,16        S  S  S  S  S  H  H  H  H  H
17,18,19,20  S  S  S  S  S  S  S  S  S  S

Soft 13,14   H  H  H  D  D  H  H  H  H  H
Soft 15,16   H  H  D  D  D  H  H  H  H  H
Soft 17      H  D  D  D  D  H  H  H  H  H
Soft 18      S  D  D  D  D  S  S  H  H  H
Soft 19,20   S  S  S  S  S  S  S  S  S  S

A-A          SP SP SP SP SP SP SP SP SP SP
2-2,3-3      SP SP SP SP SP SP H  H  H  H
4-4          H  H  H  SP SP H  H  H  H  H
5-5          D  D  D  D  D  D  D  D  H  H
6-6          SP SP SP SP SP H  H  H  H  H
7-7          SP SP SP SP SP SP H  H  H  H
8-8          SP SP SP SP SP SP SP SP SP SP
9-9          SP SP SP SP SP S  SP SP S  S
10-10        S  S  S  S  S  S  S  S  S  S

H=Hit, S=Stand, D=Double down, SP=Split

Card counting systems
A number of blackjack players have enjoyed success at beating the house by using card-counting systems. Card-counting systems are based on the principle that the make-up of a deck of cards changes as cards are played. Analysis has shown that a deck is in the player's favor when it has more ten-count cards in it. The reason the deck favors the player is because the player has a better chance of getting blackjack which pays 3 to 2, more money than an average win.

By assigning a simple value such as +1 for high cards and -1 for low cards, a card counter can roughly determine the ratio of high cards to low cards remaining in the deck at anytime during play. When the deck contains more high cards and the odds are in his favor, he increases the amount he bets so he can win more. When the deck isn't favorable, he bets the minimum so he loses less. The count also affects his decisions on when to hit, stand, double and split.

If you want to learn more about card counting, I recommend The World's Greatest Blackjack Book by Lance Humble and Million Dollar Blackjack by Ken Uston. I'm sure you can order them at Amazon.com.

Note that virtually all online casinos shuffle the cards after each round. This prevents players from using their computer to keep a count, and get an edge on the house. At online casinos, your best bet is to stick with basic blackjack strategy.