A battle (nicknamed "balle") is a short unadvertised contest where players play a new external level made by someone else for a short period of time. Statistics are made from the results and are displayed on IRC. The winner is expected to upload a replay of his best time to prove his result. The Mopobattles were a very similar concept. Those battles were bigger due to the advertising made beforehand and there were better statistics available. Nowadays, virtually all battles are played in Elma Online, since stats are created automatically there.
Before Belma and EOL, there was basically only one battle mode (best time or most apples wins) that was played, where each player played the given unforeseen level for the given time and told the best time he got. Nowadays in EOL there are several more battle types, such as first finish (first one to finish wins), one life (you can only play once) and slowness (slowest one to finish wins) among others.