If you have the body of a lion and the head of a human, you can qualify as a sphinx, but if you really want to go whole-hog, you need the wings of an eagle as well.
According to Greeks, the Sphinx was the daughter of the two-headed dog Orthus - brother of the more famous, three-headed Cerberus. (Hades could have used Orthus as a guard dog, but he was like, "Who wants to deal with a two-headed mutt?") The mother was the Chimera, an animal with a snake for a tail, a lion's head and a goat's head. The goat's head seems not so much terrifying as inconvenient. In any case, the Greeks believed that if a two-headed dog mated with a lion-goat-headed snake-tailed monster, the result would naturally be a human-headed lion, possibly with wings.
The Sphinx stood at the gates of Thebes, a city in Egypt, asking a riddle, and if you didn't know the answer, she'd strangle you. You already know the riddle and the answer because it's pretty old, but in those days, no one had heard it, and she killed a lot of people. Why someone with a lion's body would strangle people is another riddle altogether.
The name Sphinx means strangler, but it's really just an example of how the Greeks got an Egyptian myth all bungled up. Sphinx is probably a corruption of Shesepankh, which meant living image and referred to the statue of the Sphinx, specifically the Great Sphinx of Giza. Far from being a monster, she symbolized the power of the pharaohs.
That's the way it is sometimes. You create a monumental work of art, symbol of your national pride, and your neighbors turn it into a monster.