No limit Texas Hold'Em is a 7 card game typically played with 2 blinds that represent the initial bets required to start a hand. The player immediately to the left of the dealer pays the small blind which is half of the big blind, the player to her left pays the big blind which will be twice the small blind.
At this point the dealer gives each player 2 private 'hole' cards. The first player to act is to the left of the big blind, they can fold, meet the big blind or raise the bet above the big blind. Every player thereafter either folds, calls by meeting the current bet or raises until the bet comes back to the player who put in the big blind. The big blind player can bet, check (say 'check' and do nothing if there has been no raise to this point), call if there has been a raise (say 'call' and push in enough chips to meet the bet) or 'fold' to end their involvement in the hand whereupon they place their cards into the dealers discard or burn pile. Each time the bet is raised the play will rotate around to all of the other remaining players who can call, raise or fold.
The dealer then discards a card face down into the discard pile - known as burning a card - and then lays the next 3 cards face up on the table. This is known as The Flop. The player immediately to the left of the dealer (the small blind player) starts the play, either checking, raising or folding, each remaining player in turn can either check, call, fold or raise.
The dealer will then burn another card and place The Turn card face up. Another round of betting ensues. Lastly the dealer will burn one last card before turning up The River card. A final round of betting ensues. At that point the last player to bet will show their cards, building their best five cards out of a mix of the two cards in their hand and the five on the table. Other players can either fold their hands or lay them down if they either want to show their cards or believe they have a stronger hand. You must lay your cards face up on the table if you want to claim that you have won the hand.
The game proceeds hand by hand, as described above, the dealer rotating one place to the left after each hand.
The game is typically played in two ways. In casual betting games players put their money on the table and can bet as much of it as they care to. In these kinds of games players can agree to refresh their table money between hands but never in the middle of play. Tournament play has players buying a set number of chips at the beginning of the game and players are not allowed more chips during play, play continues until only one player remains. Tournament play can be played winner takes all or with declining percentages of the tournament fee going to runner up players. Single table tournament games of ten players or less can be played winner take all but are often played with second and third prizes.
Split pots: When two or more players share the same valued hand then they will split the pot evenly with any odd remainder being left in the pot for next hand.
Side pots: When a player doesn't have enough money to meet a bet then they are considered to be 'all in' and a side pot is formed for any subsequent betting. Only players who contribute to the side pot can win the side pot and if the best hand is with a player that didn't contribute to a side pot then the best hand of the contributors wins the side pot.