Evaluating your hand:
High-card points (HCP): A=4, K=3, Q=2, J=1
Long-suit points: Add 1 point for a good 5-card suit, 2 for a 6-card suit, 3 for a 7-card suit.
Short-suit points: If you have a trump fit with partner, add 1 point for a doubleton in a side suit, 2 for a singleton, 3 for a void.

Goals: Below are the bids in order. The bidding system is aimed at finding the cheapest game contracts, 3NT 4H and 4S, and for these the partnership usually needs 26+ points. For 5C or 5D, 29+ points. For small slam, 33+. A good trump fit is 8+ trumps between the two hands.
1C 1D 1H 1S 1NT 2C 2D 2H 2S 2NT 3C 3D 3H 3S 3NT 4C 4D 4H 4S 4NT 5C 5D 5H 5S 5NT 6C 6D 6H 6S 6NT 7C 7D 7H 7S 7NT

Opening bids: (making the first bid other than a pass)
1NT: Exactly 16-18 HCP, a balanced hand (4-3-3-3, 4-4-3-2, or 5-3-3-2), and no 5-card major.
1H or 1S: 13-21 points and 5+ card suit. (With two five-card suits bid the higher-ranking suit.)
1C or 1D: 13-21 points. Longest minor, but might only be 3 cards. (With 3-3 in minors bid 1C; with 4-4 or higher bid 1D.)
2NT: Exactly 21-23 HCP and balanced distribution. (With 19-20 HCP balanced, open 1C or 1D then jump to 2NT.)
2 of a suit: A very strong hand (22+ points) with a 5+ card suit. Forcing. (2NT is the negative response, showing 0-5 points.)
3 of a suit: Preemptive: a weak hand (5-9 HCP) with a decent 7-card suit. (A hand that will take 6 tricks.)
4 of a suit: Preemptive: 6-10 HCP with an 8-card suit. Don't rebid after preempting; you've described your hand to partner.

Responses to partner's 1NT opening bid: (Responses to a 2NT opening bid are the same, except 2NT 3C is Stayman)
Partner has described her hand, so you're the captain. Add up your points and guide the partnership to the right contract.
1NT Pass: Normal if game is out of reach, unless 2D 2H or 2S is a safer contract.
1NT 2C is Stayman, promising some 4+ card major and asking partner to bid her 4-card major (2H or 2S) if she has one, otherwise to bid 2D. You can subsequently shift to a major to show a 5-card suit and invite game.
1NT 2D, 2H, or 2S is a sign-off; partner must pass.
1NT 2NT is invitational to 3NT; partner will raise if at the high end of her 16-18.
1NT 3-of-a-suit is forcing. Partner will raise with 3+ card support, and otherwise bid 3NT.
1NT 3NT or 4H or 4S is a sign-off in game.
1NT 4C is Gerber, asking for aces. Partner responds 4D with 0 or 4 aces, 4H with 1, 4S with 2, 4NT with 3.
1NT 4NT is invitational to 6NT. (This is quantitative, not Blackwood.)

Responses to partner's 1-of-a-suit bid:
Pass with 0-5 points.
Simple raise (1H 2H) shows 6-10 points and support in the suit (8+ cards between you)
Jump raise (1H 3H) is forcing to game, showing 13 or more points and support.
Double-jump to game (1H 4H) is a sign-off, indicating a weak hand but with many trumps.
One over one (1H 1S) indicates 6-18 points, and 4+ cards in the suit. Forcing, i.e. don't let the bidding stop there.
Two over one (non-jump) (1H 2C) 10-18 points, and 4+ cards in the suit. Forcing (except by passed hand).
Jump shift (1H 2S) shows 19+ points. Forcing to game, and slam is likely.
1NT response (1H 1NT): 6-9 points, no fit in partner's suit, no other suit that can be bid at one level. Not necessarily balanced.
2NT response (1H 2NT) shows 13-15 points and a balanced hand, and usually no 4+ card major suits.
3NT response (1H 3NT) shows 16-18 points and a balanced hand.

Considerations for subsequent bids:
When rebidding your suit or raising partner's suit or bidding NT, you show a limit on the strength of your hand. E.g. 1H 2C 2H shows a minimum hand (13-15 points). It also shows extra length in hearts, i.e. a 6+ card suit. With a few more points, jump a level instead, e.g. 1H 2C 3H shows 16-18 points. The same principle works for responder, e.g. 1C 1H 1S 3S shows a few more points than the 6+ responder had promised earlier, and is invitational to game. Bidding a new suit is more forcing; e.g. 1S 1NT 2C might be 13-18 points, while 1S 1NT 3C is a jump shift showing 19+ points. 4NT is Blackwood (except in a NT sequence), asking for aces: 5C=0/4, 5D=1, 5H=2, 5S=3.

Overcalling after an opponent opens the bidding (and your partner passed or hasn't bid yet):
Simple overcall e.g. 1H 1S: A good 5+ card suit and 10-17 points. (With 18+, double then bid your suit.)
Jump (or multiple-jump) overcall e.g. 1H 3S: weak hand with long strong suit (like opening 3S).
1NT overcall: 16-18 HCP balanced hand with strength in opponent's suit.
Double of a natural suit bid (up to 4D): Takeout, promising 13+ points (including short-suit points) and 3+ card support in each unbid suit. Forcing. Partner normally bids her longest suit, at minimum level with 0-8 HCP, jumping a level with 9-11 HCP, and jumping to game or cue-bidding opponent's suit with 12+ HCP. (Partner can respond in NT if her best suit is the opponent's.)