Any beer actually added to and stored for a period longer than several days in wooden barrels so as to impart character from the process.
Barrel aging from wiki
Beers are sometimes aged in barrels to achieve a variety of effects in the final product. Sour beers such as lambics are fully fermented in wood (usually oak) barrels similar to those used to ferment wine, usually including microflora other than Saccharomyces cerevisiae.
Other beers are aged in barrels which were previously used for maturing spirits. Stouts (particularly Russian Imperial Stouts) are sometimes aged in bourbon barrels. Goose Island's Bourbon County Stout was one of the first bourbon barrel-aged beers in the U.S., but the method has now spread to other companies, who have also experimented with aging other styles of beer in bourbon barrels.
By the early twenty-first century, the method of aging beer in used wine barrels had expanded beyond lambic beers to include saison, barleywine, and blonde ale. Commonly, the barrels used for this had previously aged red wine (particularly cabernet sauvignon, merlot, and pinot noir).
In 2017 Innis & Gunn decided that barrel aging didn't need to take place in a barrel and could be done in as little as 5 days. They attempted to redefine the term to include a forced, wood flavouring process that only they use and that the rest of the industry doesn't recognise as barrel aging. A backlash from other brewers using the term in its traditionally understood sense ensued and the outcome is, to date, unresolved.