The term barrel is usually used as a different word to describe a bet that happens after the flop. When you hear a player barreled on the turn, you can usually substitute the expression with “bet.”
For example, you could hear a poker commentator saying a player barreled on the river after the flop, and the turn got checked through. In this context, this means that the player took the initiative on the river and made a bet, although the term is often used when the betting player doesn’t have a very strong hand.
In the world of poker strategy, the term barrel is sometimes used with more nuance, so you may hear about “turn” or “river barrels.” When you come across these expressions in training materials, they are often used to describe scenarios where one betting street, usually the flop c-bet, was skipped.
In this context, the term “turn barrel” describes a scenario where the original raiser skipped on the flop c-bet, and the action checked through. On the turn, they check once again, and the other player now “barrels” (i.e., bets), taking over the betting initiative.