Each game consists of several subsequent rounds, each with its own specifics.
Round 1: the Preflop
The first round starts with compulsory bets, the so-called "blinds"; there is a small and a big blind, and these bets are made by the two players seated to the left of the dealer respectively. This bets are required to provide the basis for the pot that the players are hoping to contest.
Blinds are put forward by players seated further in the clockwise direction from the dealer. You'll always know who the dealer is, since this player will be marked with a D-chip; this chip, in turn, moves clockwise around the table with each game, too. These compulsory bets are called "blinds" because you have to make them before you even get to find out what cards you've been dealt.
Each player is then dealt 2 cards; they are called "hole cards". Everyone considers their cards and makes a decision as to whether to play them or fold right away. Those who proceed to play take turns making their bets; you begin with the player to the left of the big blind and proceed clockwise.
This betting round is over as soon as all the players meet the highest bet put forward throughout the round; if nobody meets the highest bet (or calls the bet, which is the proper poker term for it), the one making this high bet ends up winning and taking the pot.
Now then, once this betting round is over (and there are still players participating), you move on to the next one: the flop.
Round 2: the Flop
The first three community cards are laid on the table face up. Now every player has a total of 5 cards at their disposal. This is enough to start making up hands; bear in mind that there's always a chance an even stronger hand will emerge in the next round (and not just for you, but for other players, too!).
The player to the left of the dealer is the first one to place a bet this time.
If there's more than one player "still standing" (i.e. not having folded) by the end of this betting round, the game proceeds to the next one: the turn.
Round 3: the Turn
The fourth community card is dealt face up. Players re-enter with more bets in a manner similar to the flop.
Round 4: the River
The fifth and final community card is dealt face up.
Now that all cards are revealed, players know their own best available hands for certain.
The final betting round is carried out, similarly to the flop and the turn.
Round 5: the Showdown
This is the summary of the game for all players: everyone shows their cards and the winner is determined as the player with the strongest hand. The winner takes the pot.
If there were side pots formed throughout the game (say, someone went all-in at any previous round but that didn't stop the game right then), or if there are 2 players with identical strongest hands, the pot may have to be split.