Reading the Board

It's important to be able to "read the board" and know what hands are possible and what's likely to be out there. The first part of that is identifying "the nuts". The nuts is the best possible hand. If you hold the nuts, you know you cannot lose. (Of course, the nuts on the flop may turn into a losing hand on the river). Note that you can see your hole cards, so sometimes you can have the nuts even though the board permits a better hand.  For example with AA7 on the board, AA is the best possible hole cards, but if you hold A7 you know you have the nuts, because there is only one more A in the deck.

First consider a straight flush. A straight flush is only possible if there are 3 cards of the same suit on the board which are near enough in value to make a straight. For example, a 9TJ (Nine,Ten,Jack) of Hearts on the board could make a straight flush for anyone with a QK of Hearts, or a 78, or an 8Q. Similarly, 69T makes a straight for someone with a 78, etc.

Now, for a four of a kind, there must be a pair on the board. Then, if someone holds that pair, they have four of a kind. If there are trips on the board, then someone just needs to have the one other card of that value. For example, with 33 on the board, a hole of 33 makes four of a kind. With 333 on the board, any hole 3X (like J3, 73) make four of a kind.

For a full house, you need the same thing - a pair on the board. For example, if the flop is 337, then a 37 hole or a 77 hole both make a full house. The 77 house is preferred. In this case 33 is the nuts, 77 is "second nuts", 73 is "third nuts".

For a flush, there need to be 3 of the same suit on the board. Then, anyone with 2 of that suit in the hole have a flush. If there are 4 of a suit on the board, you only need one of that suit in the hole.

For a straight, there need to be cards of nearby value. Generally on the river there's usually a straight possible if there's no pair on the board. A river like KQ269 has no straight possible.

When you have four to the flush or four to the straight, you're considered to be on a "flush draw" or "straight draw". That means you need one more card to come to make your hand. If you need two more cards to come, that's a "runner-runner" draw. Runner-runner is extremely unlikely, and you should essentially just forget about it, it will lead you to bad plays.

Let's look at a few cases :

2c 3c 4h Kd Qs

There's no flush because there's only 2 clubs. (this is called "rainbow")
There are straights, like A5 or 56 in the hole.
Top set would be KK in the hole, top two-pair would be KQ in the hole.
There's no 4 of a kind, straight-flush, or full house possible.
The nuts is "56" in the hole - the top straight.

7h 7d 9h Th As

A straight flush is possible, 8h,Jh in the hole, that's the nuts.
2nd nuts is four of a kind, 77 in the hole.
AA in the hole is the top full house, followed by TT, 99, and A7, T7, 97
A heart flush is posible for any two hearts in the hole.
Straights are 68 and 8J

In this case if you held 8hAh you have the "nut flush", and you know the straight flush is not possible.  You're
still beaten by four of a kind or any full house.

If you held A7 in the hole, the straight flush is still out, but 77 is now impossible.  You're still beaten by higher
houses like AA and TT and 99 in the hole.

The next thing to consider when reading the board is how likely it is that the better hands are out there. I'm not going to get into how to really do this correctly, that's a matter for experts. Basically you need to consider two things - 1) How many holes beat me, and 2) How likely is it that someone would have kept those cards to this point.

Now, there are 1326 possible holes (52*51/2). Once you see your own hole cards and 5 table cards, that eliminates a lot of possible holes, there are only 990 left (45*44/2). Basically you want to count how many holes make certain hands. Let's consider the 2nd example flop. There's only one hole that makes the straight flush, so all things being equal, that gives someone a 1/990 chance of having that hand. Similarly, there's just one way to have 77. To have AA there are 3 ways (3*2/2) - (AhAd) (AhAc) (AcAd). There are 45 ways to make the flush (10*9/2), that's about a 5% chance (45/990). I'm not suggesting that you actually need to do this math, just have to have some idea that there may be very few ways to make some hands, but a lot of ways to make others.

As for what's likely, basically you have to consider that people will fold hands which are junk at the time, if they have to call bets. If they can check along for free, then they'll stay with any hand. For example, consider this board :

Ac Kh 5d on the flop 6c on the turn 5c on the river

Now, if there was no betting, then people will stay along with any hand, and you have to consider that opponents many have things like any two clubs (which now make a flush).

If there was heavy betting on the flop, then people will be folding hands that don't look good on the flop. So, people are only keeping hands with A's and K's. In that case, a flush is very unlikely on the river.

One special case of this is in the pre-flop action. If there's any betting pre-flop, people are much more likely to see the flop with pocket pairs or high cards. That means hands like 37 are very rare on the flop (if there's been betting pre-flop!). Remember that all action is dictated by chips. People won't fold 37 unless they have been bet at. For example, if the big blind can just check and see the flop, he may have any 2 cards.

It's very important to think of your hand relative to the board. For example, if you have a straight, that's generally a very good hand, but if there are four cards of the same suit on the board, that's not so good! Similarly, two pair is a good hand, but if the board then pairs, that's not so good. For example :

You have TJ in the hole, a pretty decent hand
The flop comes TJA
Now you have 2 pair, that's mighty good, but QK and AJ and AT already beat you.
The turn comes K
Now, an opponent just needs a Q to have a straight, so you're not looking so good.
The river comes A

Your two pair has just disappeared! You no longer have a TTJJ hand - you play the AA on the board, and you have AAJJK as your 5-card poker hand. Your two pair is gone and it's just like you have just a pair of jacks. Furthermore, any opponent is quite likely to have an Ace, or a Queen, so you're quite likely to be beaten. You hand value looked good in the abstract, but compared to the board, it's quite poor.
A lot of beginners make the mistake of getting too attached to their hole cards. If they get AA in the hole, they're convinced they're going to win. The problem is that the game is not just a matter of who has the best hole - you have to show down a full 5 card poker hand. If the board comes with something like a 4-straight or 4-flush, or if it pairs Kings or something like that, you have to consider that you're quite likely to be beaten (depending on the betting action and opponents).

More generally, see the concept of "full hole ranking" and "bayesian poker".