Remember that the unicorn in myth has undergone a lot of changes over the centuries. Today's image of a magical horse with a horn is fairly recent (sometimes refered to as the medieval unicorn); the original descriptions were slightly different--a one-horned animal with the feet of an ox (or elephant) and the tail of a lion (or boar). Sometimes the build was described as horselike, but sometimes as oxlike or deerlike.
The myth you're refering to was that the only way to catch a unicorn was to get a virgin woman to lure it. Unicorns were innately drawn to virgins, and the woman would wait under a tree until the unicorn came to her. When it did, it would peacefully rest its head in her lap, and she would bind it with a golden cord...and that's when the hidden hunters would come out. The unicorns' horn was valued for its fabled ability to neutralize poison.
Now, the magical horse-like unicorn we think of today (the one from the above myth) almost certainly doesn't exist. But, the mythical unicorn may have actually been based on a real animal. And even if it wasn't, there have been animals that somewhat fit the description.
The biggest contender is probably Elasmotherium, an extinct species of rhinocerous. It had a single, two-meter horn on its forehead, and it had longer legs than today's rhinos, which would have given it a galloping, more horse-like gait. This description is not all that far from the original, pre-medieval image of the unicorn, so even though all evidence says that it died out many thousands of years ago, some crypotzoologists believe that a small population may have survived longer, and become the start of the unicorn myth.
Of course, there's also always mutant goats and deer and whatnot.
