A betting round is when every player that is still actively involved in the current hand is given a few options that influences the size of the pot.
The usual options during a betting round are check, raise, or fold. But once a player raises the pot, the following active players can no longer just check; they must either call, fold, or reraise (depending on the rules of the game they're playing).
In Texas Hold'em, the first betting round is before the flop. The next occurs after the flop, then one after the turn, and finally one after the river.
Example...
I couldn't believe he went through every betting round betting into the pot.