Point Count

There is a very simple (and almost universal) method of counting points. Points are used in many conventions, and can be used to discuss hands after the play. Here is the table of HCP, or high card points.
HCP
Ace4
King3
Queen2
Jack1

However, you also get points for distribution. Obviously, a hand with a distribution like 6-4-2-1 (6 cards in one suit, 4 in another, 2 in another, and 1 in another) is worth more in a suit contract than a hand with distribution like 4-3-3-3 (unless the singleton or doubleton is in the trump suit), because you can trump in the short suits in the first hand, but it will take a long time (if ever) to be able to trump in the second hand. In notrump, doubletons and singletons are very bad, because the opponents can run that suit on you. Usually, distributional points are not counted in a notrump contract. Here is the table for distribution points.
Distribution Points
Void3
Singleton2
Doubleton1

Conventions

You and your partner can communicate during the bidding about the value of your hand by using conventions. Here are some of the most common ones.

Opening Bids
Opening Bids of 1

To open one of a suit, you should have 13-19 points. You should always want to open your longest suit. If you have 2 equally long suits, you should prefer the major suits over the minor suits. If you have an equal number of spades and hearts, bid spades first (so you will be able to respond in hearts at a low level if responder does not support your bid). Some partnerships have agreed on a system known as five card majors, which means they must have a five card suit to open at the one level in a major suit. To open 1 NT, you should have 16-18 points, and fairly even distribution (no singletons, and doubletons must have a high honor). For example, the following hand:
S A 7 H K J 6 D Q 10 2 C A Q J 8 4
should be opened 1 NT, because there are 18 points, fairly even distribution, and the doubleton has a high honor.

Opening Bids of 2 or Greater

To open 2 of a suit, you should have 20 or more points, and a long suit. This is a forcing bid. To open 2 NT, you should have 22-24 points, and a stopper in almost every suit. To open 3 NT, you should have 25-27 points, and a stopper in every suit. With a very long suit (7 or more), and not enough points to open, consider opening 3 of that suit. This is called a preemptive bid. The purpose of a preemptive bid is to cut the enemy's communication so they cannot reach a contract. For example, you should open 3 spades with this hand:
S A Q J 9 7 5 3 H 5 D 8 4 3 C 8 2
To open 4 of a major suit or 5 of a minor suit, you should have 8 or 9 cards in that suit, with very few points outside that suit. For example, you would open 4 hearts with this hand:
S 2 H A K J 9 8 6 5 3 2 D J 4 C 5

Responses to Opening Bids of 1

If your partner opens, you can do several different things, depending on what he/she does and what your hand looks like. If your partner opens at the 1 level, and you have 0-5 points, pass. Even if you don't like the contract, you certainly cannot make anything higher. If you have 6-9 points, you have a few options. If you have good support for his trump suit (4 trumps or 3 trumps with an honor), raise to 2 of that suit. If you have bad support, but a suit of your own that you would like to bid, you may bid 1 of that suit, but you may not go up to the 2 level with that suit. Example: If your partner opens 1 heart, and you have good spades & 6-9 points, you may bid 1 spade. However, if you have good diamonds and 6-9 points, you may not raise to 2 diamonds. If you can do neither of these options (raising your partner to 2 or bidding your suit at the 1-level), bid 1 NT. This is not a desirable bid, but any other bid would send the wrong signal to your partner.

Responses to Opening Bids of 2 or Greater

If your partner opens 2 of a suit, you are in good shape with 6 or more points. If you have 5 or less points, bid 2 NT. This is saying that you would have passed a bid at the 1 level, but you are forced to respond. However, the most important rule when your partner opens in 2 of a suit is to never let the auction die below game, no matter how bad your hand is. If you have 6 or more points, bid your strongest suit or support your partners suit. A positive response to a opening 2 bid will always produce game, and often slam. If your partner opens 2 NT, you are not forced to respond. You should, however, respond with 4 points or more. For example, with this hand:
S Q 7 3 S 8 3 2 D Q 9 4 2 C 8 4 2
you should respond 3 NT to your partner's opening 2 NT bid. However, with this hand:
S J 7 3 H 7 6 2 D 10 5 4 3 C 8 7 4
you should pass your partner's opening 2 NT bid. If he/she could make a 3 NT alone, your partner would have opened 3 NT instead of 2 NT. If your partner opens 3 of a suit, you should generally pass. You can raise the bid to the 4 level (in a major suit only) if you have 15 or more points. If you have 15 or more points & your partner opens 3 in a minor suit, you can raise to 3 NT if your distribution is good.

Blackwood and Gerber Conventions

Normally, you can use the Blackwood convention when you are headed towards slam to ask your partner how many aces he/she has. To use this convention, you bid 4 NT. Here is the table of responses:

Blackwood Responses
ResponseNumber of Aces
5C0 or 4
5D1
5H2
5S3

However, if you are going towards a slam in notrump, a bid of 4 NT is not Blackwood! (This bid will be discussed later) If you are going towards a slam in notrump, use the Gerber convention to ask for aces. A 4C bid is Gerber; here is a table of responses:

Gerber Responses
ResponseNumber of Aces
4D0 or 4
4H1
4S2
4 NT3

Notice that, in both Gerber and Blackwood, the responses are ascending with increasing number of aces (except for 4).

Cue Bids

A cue bid is made without any desire to have the suit bid as the trump suit. This may seem pointless and needlessly complicated, but, as you will see, they can very useful. Here is an example of bidding with a cue bid:
Partner: 1H Opponent: 1S You: 2S
Your 2 bid is a cue bid. Notice that you don't really want spades to be trump - your partner can tell this since your opponent bid them! What this says is that you have a control in that suit. This also implies great support for hearts, since you didn't bother to try to find a suit of your own. Note: Cue bids are only used when there is a possibility of slam - do not use them just to tell your partner you have the ace of spades! You can continue cue bidding controls, like this:
Partner: S A J 7 5 3 H A K 5 D Q 7 C 7 5 3
You: S K Q 10 6 4 H D A K 10 4 C Q J 9 6
Partner: 1S Opponent: 2H You: 3H Opponent: pass
Partner: 3S Opponent: pass You: 4D Opponent: pass
Partner: 4H Opponent: pass You: 4S Opponent: pass
Partner: pass Opponent: pass
Whew! Let's analyze this to get a better understanding of cue bids. First, your 3H bid is a cue bid, indicating a control of hearts (a void) and shows excellent support for spades. Your partner's 3 S bid shows a control of spades (the ace). Your 4 diamond bid shows control of diamonds (the ace). Then, your partner's 4H shows control of hearts - again! Why did your partner do this? Well, he/she was letting you know that if you both have control of that suit, that his/her controls are wasted, and so a slam is probably not possible. Realizing this, you bid the contract at 4S.

Stayman Convention

The Stayman convention is a 2C bid over a 1NT bid by your partner that asks for a response in hearts or spades, whichever suit partner has four cards in. Here are some examples:
Partner: 1NT Opponent: Pass You: 2C Opponent: Pass
Partner: 2S Opponent: Pass You: 4S Opponent: Pass
Partner: Pass Opponent: Pass
For this auction, partner's hand could be this:
S Q 10 9 7 H A J 7 D K 10 3 C A Q 2
And your hand could be something like this:
S K J 5 4 H 3 2 D A Q 7 5 4 C 10 8
Notice that if you had not found the fit in spades, you could have gone to game in no-trump. But, you took the time to see if partner had any four-card suits, and since partner had spades, you decided to go for the easier 4S game. Here's another example:
Partner: 1NT Opponent: Pass You: 2C Opponent: Pass
Partner: 2C Opponent: Pass You: 3NT Opponent: Pass
Partner: Pass Opponent: Pass
In this auction, the 2D response to the Stayman convention indicates partner doesn't have 4 cards in hearts or spades. Since there is no easy major suit game, you go to the no-trump game.

Takeout Double

A takeout double is used to convey a message to your partner that is quite different from a penalty double. To be a takeout double, two conditions must be met, or else it should be considered a penalty double.
When you double, you are asking your partner to bid his/her best suit (obviously not what the opponents bid). The fewer cards you have in the suit you doubled over, the better. This may sound kind of abstract, so let's go over it with a few examples. In both of these examples, your LHO just bid 1H.
You: S 10 8 7 5 H Q J 9 8 6 4 D A C A 3
You: S A Q 8 6 H 5 D A J 7 4 C Q 9 8 6
If you takeout doubled with the first hand above, partner would likely bid diamonds (he/she is likely to have more cards in the suits you have less of), and you would be in trouble, for you have no idea what suit to play in. So, you would probably pass. However, a takeout double is very good with the second hand, because you can support any suit that your partner bids. Note that you do NOT say "takeout double", you can only say "double", and your partner must learn the difference between a takeout and penalty double.

Overcalls

An overcall is something that is done when your opponent opens. For example, if your LHO opens 1D, and you bid 1S, this is an overcall. Responding to an opening bid from your partner is generally safe, because you know that he/she has at least 13 points, so you can probably make a 1 or 2 level contract. However, it can be dangerous to bid after your opponent opens, for your side may be stuck in an unmakable contract if your partner has very little. So, in order to overcall, you must have a strong 5 card suit if the overcall is at the one level, and a strong 6 card suit if the overcall must be made at the two level.