The most popular kind of poker today is the one I described, Hold Em. Otherwise known as Texas Hold Em, though it's played all around the world now. The classic kind that you're familiar with is called 5 Card Draw, you get a hand of 5 cards, and instead of community cards you can discard and draw up to 4 new cards. The other most popular form of poker is 7 Card Stud, in which you get 3 cards down and 4 cards face up, but the cards are all yours unlike Hold Em where the face up cards are shared by everyone.
Betting is pretty straightforward. If no one has bet yet, you can either bet or check/pass. Checking simply means you decline to make a bet, but you are still in the hand. If someone has already bet, then you can either fold, call, or raise. Folding means you refuse to match the bet, but you have to forfeit your hand and can no longer win. Calling means you match the bet and are still in the hand. Raising means you increase the size of the bet.
"Betting thin" is an advanced term in current poker theory, it means to bet for value with a relatively weaker hand. Betting for value means you expect to win more often than not if you get called, hence "value." So to "bet thin" means you are close to the hypothetical border to where you can still expect to win if you are called. Advanced poker theory in general is about getting the maximum "expected value" out of all of your actions... usually there are 3 different actions you can take every time it's your turn, fold/call/raise, and each one has a theoretical expected value, advanced poker theory is about figuring out the value of each play, and then making the best one.