Limit

In limit Texas Hold'em you are limited to have much you can bet when it is your turn. The amount you can bet or raise is predetermined. Casinos offer limit games ranging from $1-2 to $100-200 or even higher like $4000-8000. Online cardrooms offer games as low as 1cent-2cent. The limit of the game determines the amount that can be bet. In a $1-2 game, the bets will be $1 and $2. In a $10-20 game, the bets will be $10 and $20. Before the flop and on the flop, the bet/raise amount is called the small bet. On the turn and the river, the bet/raise increment is the large bet. For example, if you are playing in a $1-2 game, you can bet or raise $1 at a time before the flop and on the flop. On the turn and the river you can bet or raise in increments of $2.

No-Limit

In No-Limit you can bet or raise as much as you like - but always at least as much as previous bet or big blind. That is, as much as you like of the money you have on the table when the hand is started (called table-stakes). When you bet all the money you have left you are all-in and can make no further decisions in the hand.

Tournaments

Tournaments can be played with any betting structure, but is most commonly played as No-limit. They are normally scheduled to start at a certain time. In tournaments you buy in for a certain amount and then you get a bunch of chips to play with. For example, you can buy in for $10 and get 1500 chips. When you are out of chips you are out of the tournament. The bet sizes are increased after certain periods of time - so it will not last forever. The better you place in a tourmanent the more money you will win. Usually the winner of a tournament gets a large part of the prize money. A prize structure determines how much different placings are worth.

Sit & Go

A Sit & Go is a special kind of tournament that starts when all seats has been filled. The most common Sit & Go format is for 9 players. You can sign-up for 9 player Sit & Gos at any major poker room and wait a couple of minutes to get started.

Other variants

The above are the most common variants and enough to get you started. Some other variants that might be good to know of are:
Pot-Limit: You can only raise with the size of the pot.
Spread-Limit: Restricted like Limit, but you can choose within those limits.
Rebuy tournaments: If you loose all your chips you can rebuy and continue playing.
Turbo tournaments: The number of chips you start with is often smaller compared to the blinds and the size of the blinds increase faster than in a regular tournament.