No-limit Texas Hold'em Rules
Texas no-limit hold'em is the world's most popular variant of poker.
Simple Overview
The dealer shuffles a standard 52 card deck, and deals two cards face down to each player - these first two cards are called 'hole cards', and are known only to each player.
Throughout the game the players will have several rounds of betting during which 5 more cards will be dealt face up (called the 'board'). If a player thinks that their hand isn't strong enough to bet with, they may fold it and be out of the game (forfeiting all money they have bet up to that point)
A player must make a 'hand' using any 5 cards from the 7 available (ie both of their hole cards plus 3 from the board, or just 1 of their hold cards and 4 from the board. If both their hole cards don't contribute to a decent card, then they may be playing all 5 cards from the board).
The categories of hands can be seen here.
At the end of the game the hands are shown ('showdown') and the best hand wins the money that was bet. Often however all but one of the players has folded their cards, and the game doesn't reach showdown.
Detailed Description
A marker called the dealer's button is placed in front of a player. After each hand the button moves clockwise round the table. The button defines the position of each player, with play always starting on the button's left, moving clockwise round to the button (which means the button is always the last person to act).
The hand then starts by the blinds being posted by the players sitting to the left of the button - the blinds are small bets that determine the minimum size of betting for the rest of the hand. There are two blinds, the small blind and the big blind. The small blind is usually half the size of the big blind and is posted by the player immediately to the left of the button (and so the position of that player is called the 'small blind' or SB). The big blind is posted by the player to the SB's left (and accordingly that position is called the 'big blind' or BB).
The dealer (or the player on the button), deals two cards face down to each player.
The player to the left of the BB (called 'under the gun' or UTG) starts the first betting round. The UTG has three options - to fold his hand if he feels it's not worth playing, 'calling' by matching the big blind (ie putting in the same amount as the big blind into the pot), or 'raising' by putting more than the big blind into the pot. If the UTG raises the raise must be at least as large as the last bet (ie the blind is $5, then the UTG either calls by putting $5 in, or raising at least $5 and putting $10 in).
Moving round the table clockwise each player decides whether to fold, call or raise. This will continue round the table until everyone has either called or folded.
Here's an example;

6 players are in the game. The stakes of the game are $1/$2 (which means the small blind is $1 and the big blind is $2).
SB puts in $1.
BB puts in $2.
The dealer gives each player two cards face down.
The UTG has 4
8
and decides that he won't win much so he folds.
The MP ('Midde Position') has 9
T
and decides to call - he puts $2 into the pot.
The CO ('Cut Off') has 2
J
decides to fold.
The BTN ('Button') has 6
6
and also calls, putting $2 into the pot.
The betting has gone all the way round the table (the blinds are 'forced' bets essentially), but since the SB's bet was only half the current bet he still has to call the remaining half (or raise or fold of course). If the SB does call, the BB has essentially already made his bet, and can just 'check' to stay in the hand
The BB has A
A
and decides to raise - he raises $4 and puts $6 in.
The SB has 7
3
and folds.
Since there was a raise, the betting continues round the table until everyone has called or folded
The MP considers folding, but decides to call instead. He puts the $4 to match the raise, bringing his bet up to $6.
The BTN decides to fold.
That concludes this round (or 'street') of betting.
Once that betting round is over, the dealer places three cards face up in the middle of the table - this is known as the flop. The cards that are face up are 'community cards', ie shared between all the players. The face up cards are known as the 'board'. All the players now have a much better idea of how strong their hands are. A new round of betting starts, beginning with the SB.
After the flop there are two more streets - the turn and the river. For both these rounds the dealer places a single card face by the community cards, followed by another round of betting.
If the hand reaches the final street then after the final round of betting the cards are shown and the best hand wins the pot. It is possible that the final hands have the same strength, and in the case of a tie the pot is shared equally.
Side Pots
Each player starts with a set amount of money called their 'stack'. Games may have set limits to the amount players sit down to play with (either a maximum or minimum stack). As the game progresses and the money moves from player to player the stacks will begin to vary in size.
The difference in stack sizes may create a situation where a side pot has to be created. The easiest way to explain this is with an example;
3 players are left in the hand;
UTG: Stack of $30
CO: Stack of $5
BTN: Stack of $50
The UTG makes a bet of $20. The CO hasn't got $20, but he has $5 and is allowed to put the remainder of his stack in as a bet. He does so, and is 'all-in'. If the CO wins the hand he can't take the whole pot as he only contributed $5. So the main pot is capped at the amount his all-in matched (ie $10 - $5 from the UTG and his $5 call). The remainder (the rest of the UTG's bet - $15) becomes a side pot.
The bet is still $20, and the BTN calls. $5 goes into the main pot, and $15 into the side pot.
If the CO wins the hand, he takes the main pot. That leaves the side pot - this is decided by who has the best hand out of the BTN and UTG.