The Octomammoth was once an ordinary octopus that was mutated heavily by unspecified "underwater experiments." Tellah comments that it is "an octomammoth," indicating that it is not the only mutation of its type. Though no others appear in the game or its sequel, Final Fantasy IV: The After Years, two very similar monsters appear, Octo-Kraken and Cooler Mammoth. Palom mentions that the Cooler Mammoth once gave Tellah trouble, indicating it may be the same creature.
In all releases of Final Fantasy IV besides the DS release, Octomammoth is very weak. It only uses physical attacks during the entire battle, none of which are very strong. Cecil should attack while Tellah and Rydia cast Thunder. Octomammoth should perish with minimal difficulties.
The Octomammoth can be rough if the player has not been battling along the way. Cecil should have collected the Shadowblade and some Hades-type armor and use Darkness, then use physical attacks with Darkness status. Tellah and Rydia should use Thunder, and occasionally, if they have enough MP, summonChocobo with Rydia. The party should heal with Cure or, hopefully, Cura when Cecil and the others get weak. After losing a percentage of HP, the beast's tentacles will vanish, and when it has about four left, Tellah will warn that a wounded beast is dangerous. After this, the Octomammoth will slow down, but deliver massive hits that could possibly defeat Tellah or Rydia in one hit, and leave Cecil weak.