An event that was very, very popular in the big band and early television days.
The gist was that you'd have two (sometimes three) well known drummers on a stage, and they'd in essence "compete" with drum solos one against the other. They'd start out playing the same beat in unison, then one would take four or so bars to play a riff, the other would answer in a call/response manner while the original soloist returned to playing the rhythm, and so forth.
Buddy Rich was called upon to do a lot of these and some with Gene Krupa or Louie Bellson were recorded. At one point Buddy did one with Jerry Lewis, a not incompetent drummer himself, but Lewis' contribution was mostly comic.
Some of these would start out as literal call/response, in which one would play a drum fill as a challenge for the other to copy, and the other would return the favor, sometimes adding a flourish or better spin on it at the end if he could.
After a few iterations of this one would take 16 or so bars to really go to town, showcasing technique, speed, style, and technical complexity as well as showmanship, then ceding the spotlight for 16 bars for the other to answer.
Over time it fell off as a late show or band leader gimmick, but Johnny Carson and other variety shows later on in television history tried it again. The Muppet Show famously had Animal go against Buddy Rich.
And Dave Grohl.
Of course we have the joke ones, like Will Ferrell against the Red Hot Chili Peppers' Chad Smith based on a weak joke that they looked similar.
The point wasn't so much the event as a competition with a "winner" or "loser" - obviously people would have an opinion on who did the better job, but there was no formal prize. The point behind it was to give the drummer a chance to show that he or she was capable of far more than the simple, time keeping job a drummer usually has to endure.
That being said, the concept did get expanded to actual, formal competitions, but these are usually characterized as "drum-offs" as opposed to the drum battle format. In these, round-robin style, anyone who wishes to compete does so, and the winners move on, but they're judged in isolation with their five minutes of solo time as opposed to an event where they drum together and trade-off. Guitar Center has one of the largest of these in the US.