Let no joyful voice be heard
A Leviathan was originally a humongous sea beast created by god to feed the men when the end times came. It was created with a mate, who was later killed since nothing would survive a world filled with Leviathans. Since then, the term has been broadened to any massive sea creature in mythology and fantasy.
Let no man look to the sky with hope
Sea Dragons, a very popular version of leviathan, are just that: Dragons who live in the sea. Generally wingless, and a lot of the times limbless, Sea Dragons are the more aggressive kinds. Normally not breathing fire, Sea Dragons often will rather shoot boiling water, or have some kind of control of the tides or weather.
And let this day be cursed
Another popular incarnation of the Leviathan, is just like any other animal, just needlessly big. Whether it is a predator or filter feeder, often the stories told of these beasts are just accidents with no intent to kill or defend its territory, but rather it didn't see us because of its sheer size
By we who ready to wake...
Yet another idea on the Leviathan, is a Cthulhu-esque humanoid sea creature. Much like Urusula at the end of the little mermaid (except, usually not beaten by a boat). These are often seen as god or godlike creatures, with some sort of influence on the ocean.
The Kraken, of seafolk lore, is essentially a massive squid or octopus, who preys on pirates and sailors alike. Its popularity skyrocketed with the release of Pirates of the Caribbean: Dead Man's Chest and its theme is the only song on piano I have gone out of the way to learn. In Pirates of the Caribbean Mythos, the Kraken is Davy Jones' pet. This massive creature is designed to shred and crumple ships that it has been ordered to destroy, leaving no survivors.