Many mythical beasts seem to be based on real animals, just hybridized and altered in various ways. While some, like the unicorn, are well-known, others get somewhat less attention. While this is far from a complete list, as every culture has many creatures like this, here are some of the best known mythical creatures with no human (or totally alien) component to their construction.
The griffin (spelled elsewhere as griffon or gryphon) was thought to be a creature with the body of a lion together with the head and wings of an eagle. As the lion and the eagle were thought to be rulers of their respective domains - the lion ruling over the beasts of the land, the eagle ruling over the air - this combination was considered to be regal and powerful, and sometimes described as 'the king of creatures'. In addition, it was sometimes connected to Jesus, as it was a fusion of the terrestrial and the aerial just as Jesus was a combination of the earthly and the divine. Its claws and feathers were thought to have healing properties, and cups formed from antelope horn were often claimed to be formed from griffin claws. In heraldic imagery, the union often indicated a fusion of intelligence and strength into a harmonious whole.
The chimera, originating in Greek myth, is unique if only for the sheer amount of animals involved in its body. The original version had the body of a lion, a tail ending in the head of a snake, and a third head around the middle of its spine resembling that of a goat. The creature was also able to breathe fire. Many other variations exist, some setting all three heads in the forefront of the body, some adding other creatures or traits. The term "chimera" is now used for many creatures of this sort of hybridized nature.
Cerberus is another well-known creature of Greek myth, having appeared in many popular works. Cerberus takes the appearance of a massive three-headed dog, although some works reduce the number of heads to as low as one or increase the count all the way up to fifty. Cerberus was a guard at the gates of the Underworld who prevented the souls of the dead from escaping. One of the labors Hercules was forced to undertake involved capturing this beast alive, which he succeeded in by subduing the beast and carrying it out. It still remained a fearful creature.
The basilisk is another creature that's enjoyed a recent surge in popularity due to fiction works like Harry Potter. The basilisk was said to be a monstrous serpent or lizard that was born by allowing a chicken to incubate the egg of a snake. (Interestingly, a creature with similar abilities, the cockatrice, was said to be created when a snake hatched the egg of a chicken.) The basilisk had a gaze that killed anyone that it looked upon, and was said to have a poison that would actually travel up any object it was touched with and kill the holder as well. Luckily, it did have vulnerabilities - it was said to be defeated by just the smell of a weasel, and it disliked the crow of the rooster. In addition, it was once defeated when a man forced it to look into a mirror. It was considered a scourge.
As can be seen, many mystical creatures have their roots in real animals, but with hybridization or extra traits making them distinct. Many more examples of this exist - the hydra, the dragon, and of course many fusions of animals and men - but this is a good introduction to the many mystical beasts out there.