The poker term barrel is generally speaking synonymous with the term bet. In other words, a player who is barreling is making a bet.
More specifically, however, the term barrel applies to bets across multiple streets. For example, a player betting on the flop and then continuing to bet on the turn will often be described as barreling, i.e., firing out multiple bets.
The term has become popular in recent years with the growth of various poker training sites, as it is very convenient to describe spots where players take aggressive routes.