Counting Outs
Simply put, an out is a card that can land on the turn or the river to complete your drawing hand. Players need to be able to count their outs in order to better decide whether or not to pay to see the next card in a hand.
There are some common "outs" totals that are worth memorizing.
Flush Draw - if you have four of a suit, and need to complete a flush, there are 9 cards of that suit that are unseen. So, to complete your flush you have 9 outs.
Open Ended Straight Draw -
if you sit with four cards in sequence, and you can complete the straight by landing a card on either the top or bottom end of the sequence, there are 8 outs available. For instance you hold 4,5,6,7. You can complete the hand with either a 3 or an 8.
Double Gutshot Straight Draw - If you hold a series of cards like 3,5,6,7,9 you can fill in the straight with either a 4 or an 8. In this case you have 8 outs.
Gutshot Straight - drawing to a straight where there is only one rank of card that can complete it leaves you 4 outs. This is true for cards like 4,5,7,8 and also for single ended stragiht possiblities like A,K,Q,J.
Deceptive Outs
Those common totals can be deceiving. One thing to look for when computing outs is the chance that you hand will be the best hand at the table should you catch an out. This is not always the case.
One example is when you are drawing to a straight, but there is a flush draw on the board as well. There is a chance that you will land your card, but its suit will also fill in a possible flush. Another example is when you are drawing to a non-nut flush. An opponent in that situation can be drawing to a better flush, so you can once again catch your "out" and still lose the hand.
In order to deal with these situations properly, you will need to discount, or disqualify the value of some outs.
Example 1:
You are waiting for the river card to come, and have an open-ended straight (7,8,9,T). There are four players still in the hand, and a flush draw is on the board (clubs).
If you feel strongly that one of the remaining players is on that flush draw, you may decide to calculate your pot odds using 6 outs instead of 8 because if the 6 of clubs or the J of clubs lands, you feel that the pot will belong to your opponent who lands their flush draw.
There are plenty of times you will be drawing to a non-nut hand. A king high flush is a good example of that. In those instances, you might want to discount your outs in order to compensate for the times when you hit your out, but lose the pot.
Hidden Outs
There are a few more instances where you can adjust the number of outs when doing your pot odds. One of them is when you have a backdoor draw. Backdoor draws are weak, but they do exist as anyone who has even fallen prey to a runner/runner draw will attest.
Sklansky has written some analysis of backdoor draw situations, and you can read them in detail in "Small Stakes Hold'em". In short, he recommends attributing roughly 1.5 outs for each backdoor nut draw you have. If you are sitting on an Ace high backdoor nut flush draw, you may want to keep that in mind.
I don't recommend worrying about the following things until you are very comfortable at counting outs and figuring your pot odds. I mention it for the sake of completeness more than anything else.
There are other instances of hidden outs, like when the river card may cause a split pot by counterfeiting a part of your opponent's hand.
For instance, you hold ATo and your opponent holds AJo. The board is A,9,2,2. Right now your opponent is in the lead with his better kicker. But, if an Ace, a nine, a two, a King, a Queen, or a Ten come on the river, either you win or the pot splits.
This is a circumstance in which finding the outs is tricky, but expert players can see them. That is part of what makes them experts. And, you should be aware that situations like this exist. Eventually, you will be able to see the outs in many places where they were once hidden.