In order to win, the player must remember which cards have gone. This gives him the number of remaining cards in the shoe. Players with a good memory can do that easily, since they have to remember just 10 independant values. The bigger problem is how to exploit the knowledge about the cards distribution in the shoe.
Generally speaking, the low valued cards 2, 3, 4, 5 and 6 help the player whereas high valued cards 10, J, Q, K and A help the dealer. This is pretty obvious because a card valued 5 helps the dealer to bring a stiff to a final score of 17 to 21. A stiff is a score of 12 to 16 and here the dealer is risking to bust his hand. This chart also shows, that the more high valued cards are in the shoe, the better it is for the player, whereas the more low valued cards are in the shoe, the better it is for the dealer. Depending on which cards are gone in the previous rounds, the total expectation varies between about -8% and +8%.
By removing a card from the shoe, the total expectation varies a little. Removing a card with a value from 2 through 8 increases the total expectation, whereas removing a card with a value of 9, 10, J, Q, K and Ace decreases the total expectation. Since the variation of the total expectation is the key to success when attempting to beat the dealer, it is fundamentally important to compute this value out of the current distribution.
This isn't a simple task, specially if one has to do it by mind. Therefore some people have invented simplifications. One of those simplifications is named the "High-Low" counting method. Here the players assigns +1 to the low valued cards (2 to 6), -1 to the high valued cards (X and A) and 0 to the rest (7, 8 and 9). A player can easily sum up and remember this count value. Now he has to divide this sum through a certain factor depending on the number of decks during reshuffle. With this method the player can estimate if the total expectation is above zero, in which he shall place a higher stake.
The problem with all counting methods however is, that the total expectation does not vary linearly with the remaing cards in the shoe. All counting methods assume that the total expectation is a number which can be computed by a linear equation. But as we have seen, computing the total expectation is a far more complicated task, which can only be solved exactly through algorithmic analysis. For instance, take out all cards with a value of 10. The linear equation would give you a total expectation of -6.72% but algorithmic analysis computes +0.05% as the total expectation.
This shoe with two deks of cards has been just reshuffled and no card was removed yet. Look at how much the total expectation varies, if the player removes one card of a certain value.
Look what happens if some cards have been removed from the shoe:
Out of this shoe with two deks, many of the low valued cards have been removed. A shoe with this distribution of cards has a total expectation of phantastic +7.1%. Gamlers say, "the shoe is hot" and bet with a high stakes. Compare the delta (ΔtE) for each removed card with those from the first chart.
Let's look at another extreme situation:
Out of this shoe with two desks, many of the high valued cards have been removed. A shoe with this distribution of cards has a total expectation of miserable -7.1%, thus a player shall leave the table and wait for better circumstances. Compare the delta (ΔtE) for each removed card with those from the first and second chart. In some circumstances, removing a 9, 10, Jack, Queen or King reduces the total expectation, in others it increases the total expectation.
A counting method such as "Hi-Low" can only be considered as rough approximation, to determine the total expectation. This method is easy to remember, legal to apply and deviations from the real numbers occur to a lesser extend, so is is perfectly suitable for long termed gaming sessions.
Since many casinos introduced shoes with integrated card shuffeling systems, all card-counting schemes are useless nowadays. In those casinos the total expectation is uniformly around -0.85%, which is still better than playing roulette, but only if you play according to the correct strategy.