Leviathan's physical attack is Water-elemental, and may also cause Instant Death. Leviathan is weak to Fire.
Initially, Leviathan only has one arm. When it is "broken" with sufficient damage, a second arm will join it, repeating until there are four arms. When all four arms take sufficient damage, Leviathan flips over into its true form. At this point, it can attack up to five times per turn (once per arm and the main body). Leviathan will also begin casting magic at this point. It will also start charging Tidal Wave, but doing a significant amount of damage will break this attack.
Leviathan can be difficult because of its frequent attacks and chance of Instant Death. A White Mage may be advisable over a Salve-Maker due to the greater healing of Raise over Phoenix Downs and being able to re-cast Lux more frequently.