A battle for survival is a Boss fight where an individual player's survival is the most important aspect.
Some fights require all raid members to learn how to evade damage,and/or force the healers to keep the whole raid up. Typically such fights have long enrage timers (or none at all), and sometimes random or hard to control aggro. They are the most difficult type of encounter, because all raid members must learn something new. Battles for survival remain a challenge even for well-geared groups, although good gear usually allows for a larger margin of error.
Some players consider battles for survival to be the most interesting type of fight, others hate them because they are inherently more difficult. Usually, they are found near the end of instances, as most end bosses have at least one phase which is designed as a battle for survival.