A soulmate is somebody with whom one has a feeling of deep and natural affinity, love, intimacy, sexuality, spirituality, and/or compatibility. A related concept is that of the twin flame or twin soul – which is thought to be the ultimate soulmate, the one and only other half of one's soul, for which all souls are driven to find and join. However, not everyone who uses these terms intends them to carry such mystical connotations.
One theory of soulmates, presented by Aristophanes in Plato's Symposium, is that humans originally consisted of four arms, four legs, and a single head made of two faces, but Zeus feared their power and split them all in half, condemning them to spend their lives searching for the other half to complete them:
"[Primeval man] could walk upright as men now do, backwards or forwards as he pleased, and he could also roll over and over at a great pace, turning on his four hands and four feet, eight in all, like tumblers going over and over with their legs in the air; this was when he wanted to run fast …Terrible was their might and strength, and the thoughts of their hearts were great, and they made an attack upon the gods ... Doubt reigned in the celestial councils. Should they kill them and annihilate the race with thunderbolts, as they had done the giants, then there would be an end of the sacrifices and worship which men offered to them; but, on the other hand, the gods could not suffer their insolence to be unrestrained. At last, after a good deal of reflection, Zeus discovered a way. He said: 'Methinks I have a plan which will humble their pride and improve their manners; men shall continue to exist, but I will cut them in two and then they will be diminished in strength and increased in numbers; this will have the advantage of making them more profitable to us. They shall walk upright on two legs, and if they continue insolent and will not be quiet, I will split them again and they shall hop about on a single leg." —Aristophanes, Plato’s Symposium, 
Some people believe that souls are literally made and/or fated to be the mates of each other, or to play certain other important roles in each other's lives.
According to theories popularized by Theosophy and in a modified form by Edgar Cayce, God created androgynous souls, equally male and female. Later theories postulate that the souls split into separate genders, perhaps because they incurred karma while playing around on the Earth, or "separation from God". Over countless reincarnations, each half seeks the other. When all karmic debt is purged, the two will fuse back together and return to the ultimate.