What exactly the Behemoth and Leviathan are is still an open question. No one is sure what they are, so the word used for them was left untranslated. However, what is important is that they are symbol of uncontrollable natural power. The point is that man cannot subdue them because we are not in control of nature. If we can't even take on God's creatures, then we can't take on God, who is infinitely more powerful than his creatures.
However, discussing what they could possibly be is fun, if kept in perspective. The three general ideas is that they are mythical creatures, dinosaurs, or large normal animal that could still be around today. The idea of mythical creatures is also split between people who believe mythical creatures were real at the time and those who think that mythical creatures are being references just to make a point.
Behemoth (Job 40:15-24)
Behold, Behemoth, which I made as I made you; he eats grass like an ox. Behold, his strength in his loins, and his power in the muscles of his belly. He makes his tail stiff like a cedar; the sinews of his thighs are knit together. His bones are tubes of bronze, his limbs like bars of iron.
“He is the first of the works of God; let him who made him bring near his sword! For the mountains yield food for him where all the wild beasts play. Under the lotus plants he lies, in the shelter of the reeds and in the marsh. For his shade the lotus trees cover him; the willows of the brook surround him. Behold, if the river is turbulent he is not frightened; he is confident though Jordan rushes against his mouth. Can one take him by his eyes, or pierce his nose with a snare?
Some believe it is a dinosaur, like a sauropod, due to the description of the tail), or even a triceratops. Many think this is either an Elephant (the largest land animal) or a Hippo (because it lives near and goes in rivers. However, neither of these really has a tail like a ceder.
Leviathan (Job 41:1-34, Isaiah 27:1)
Can you draw out Leviathan with a fishhook or press down his tongue with a cord? Can you put a rope in his nose or pierce his jaw with a hook? Will he make many pleas to you? Will he speak to you soft words? Will he make a covenant with you to take him for your servant forever? Will you play with him as with a bird, or will you put him on a leash for your girls? Will traders bargain over him? Will they divide him up among the merchants? Can you fill his skin with harpoons or his head with fishing spears? Lay your hands on him; remember the battle—you will not do it again! Behold, the hope of a man is false; he is laid low even at the sight of him. No one is so fierce that he dares to stir him up. Who then is he who can stand before me? Who has first given to me, that I should repay him? Whatever is under the whole heaven is mine.
“I will not keep silence concerning his limbs, or his mighty strength, or his goodly frame. Who can strip off his outer garment? Who would come near him with a bridle? Who can open the doors of his face. Around his teeth is terror. His back is made of rows of shields, shut up closely as with a seal. One is so near to another that no air can come between them. They are joined one to another; they clasp each other and cannot be separated. His sneezings flash forth light, and his eyes are like the eyelids of the dawn. Out of his mouth go flaming torches; sparks of fire leap forth. Out of his nostrils comes forth smoke, as from a boiling pot and burning rushes. His breath kindles coals, and a flame comes forth from his mouth. In his neck abides strength, and terror dances before him. The folds of his flesh stick together, firmly cast on him and immovable. His heart is hard as a stone, hard as the lower millstone. When he raises himself up the mighty are afraid; at the crashing they are beside themselves. Though the sword reaches him, it does not avail, nor the spear, the dart, or the javelin. He counts iron as straw, and bronze as rotten wood. The arrow cannot make him flee; for him sling stones are turned to stubble. Clubs are counted as stubble; he laughs at the rattle of javelins. His underparts are like sharp potsherds; he spreads himself like a threshing sledge on the mire. He makes the deep boil like a pot; he makes the sea like a pot of ointment. Behind him he leaves a shining wake; one would think the deep to be white-haired. On earth there is not his like, a creature without fear. He sees everything that is high; he is king over all the sons of pride.”...
In that day the Lord with his hard and great and strong sword will punish Leviathan the fleeing serpent, Leviathan the twisting serpent, and he will slay the dragon that is in the sea.
Sounds like a water dragon (due to the smoke and fire), some believe dinosaur like a Plesiosaur or Mosasaur. The most common real animal is a crocodile, but that does not explain the smoke and it is not really a marine animal. Others think it is a whale (which is the largest marine animal) and that the water coming out of the blow hole was mistaken for smoke. This does not explain the sharp scales, but maybe it is barnacles all over them. Whales are also not very serpent-like.
Ziz (Psalms 50:11, 80:13, 2 Chronicles 20:16)
I know all the birds of the hills, and all that moves in the field is mine.
The boar from the forest ravages it, and all that move in the field feed on it.
Another so-called mythical creature, supposedly from the Bible is the Ziz. This is a transliteration from a Hebrew word that is commonly translated as, "all that moves." It is an ancient Hebrew mythological flying creature that some say is a magical bird like a Anzu or Phoenix, or other beast like a Griffin. In this thought, it is the flying (air) counterpart to the walking Behemoth (land) and swimming Leviathan (sea) creatures. Some say it is a dinosaur like a Pteranadon or Scaphognathus. Lastly others say it is a large bird like ostrich or emu (because it runs around in the fields) or a large scavenger flying bird like a condor. Most likely these verses are not actually referencing a specific animal.