The Lizard Elder is a neutral monster in League of Legends described in-game as, "a melee monster with large amounts of health. Killing this foe grants a slowing attack that deals damage over time." It only appears on Summoner's Rift, where it spawns with two Young Lizards.
When killed, the Lizard Elder grants its killer the Blessing of the Lizard Elder buff (very often just referred to as Red buff), where every hit deals 10-44 (8 + [2 x level]) bonus true damage twice and slows enemies (8/16/24% for melee or 5/10/15% for ranged attacks) for 3 seconds.
Starting patch 3.14:
Jungle monsters now have levels
Jungle monster level is based on the average level of the champions in the game when they spawn (equal to the average of all champion levels, rounded up)
Jungle monster level determines their HP / Damage and XP / Gold value
Controlling the Lizard buffs can give your team an advantage in the match. Most junglers will attempt to control not only their own but also the opposing team's buff.
The lizard spawn is indicated by a gold marker on the minimap. If your team has vision when or after the lizard is killed, the gold marker will disappear and reappear when the camp respawns.
The lizard will respawn regardless of whether its buff is still on the map, so there can potentially be more than two of the buffs active if their durations have been reset for long enough. The same applies for the Ancient Golem.
The lizard does not respawn if the young lizards are not killed. This can be used against the enemy when taking buffs from the enemy jungle, as it robs them of potential buffs until they kill the remaining creeps and it respawns.