Māra comes from the root mr which means die or dying. So Māra is the god (deva) of Death or the Dying God (or both). His heaven world is the highest of those devoted to sensual pleasures. Like all other gods, he is not immortal, though his lifespan is inconceivably long compared to that of a man. His heaven world is filled with those who have devoted their lives to sensual pleasures (see the sutta about Māra's Net). But those who pursue indulgence in sense pleasures (including Māra himself), have to be born into a world (heavenly or otherwise) where sense objects are available in order to enjoy them. Being born, they will have to die.
This is the flaw in Māra's world, his followers and his own being. He devotes himself to activities which everywhere and always result in death. Escape from death is only to be found by rejecting the pursuit of sense pleasures and therefore not being born.
When Māra realised that the Buddha was attempting to find this escape route from birth and death into the deathless (amara = a + māra), he tried to prevent him. He saw that, if beings responded to the Buddha's teaching, they would escape from his net and he would lose his power both here and. in his heaven world.
So he sent his daughters, Tanhā, Arati and Rāga, to tempt the Buddha back into the plane of being where the pursuit of sense pleasures is the goal.
Tanhā means compulsive thirst, like that of an alcoholic for alcohol; and therefore craving. There is sensual craving, craving for existence and craving for non-existence. Sensual craving is craving for the six sense objects. Craving for existence is craving for the world of the senses or the world of form or the formless world. Craving for non-existence is craving for extermination.
Rāga means sensual desire.
Arati (a + rati) is the opposite of Rāga and therefore means aversion. Rati [Classic Sk. rati, fr. ram] means love, attachment, pleasure. As a proper name, Rati is the goddess of lust and pleasure.
Māra's daughters, therefore, represent those things which stand between us and the goal of amara.
How did the Buddha overcome them?
NEW PROJECT: THE DOG WITH MANGE
Note: "During the years when the Lord Buddha was alive, he told a story of a dog with mange. When it stayed in a cave it thought that the cave was uncomfortable and the dog itched all over. It moved to a forest and still it itched and was uncomfortable. It was never comfortable no matter where it went. This is because what was causing the itching was its skin condition, not the place it stayed."