We don't really know - the unicorn doesn't appear in actual mythology, because people genuinely believed they existed. In fact, the earliest unicorns weren't even horses with horns, but were more like large goats. Narwhal tusks supported the legend until people realized they came from a kind of dolphin.
Another version described unicorns as being akin to wild asses. Rhinos, of course, have central horns, and oryxes often look like they do unless seen from the front (many think the oryx is the actual creature that started the story). Marco Polo described the unicorns he saw on his travels - and he had clearly seen a rhino.
Unicorns in fantasy may be horned horses, but the unicorn of legend is definitely something else. No equine species has had horns as a routine thing, although they do show up as an extremely rare mutation. Most likely the original unicorns were, then, just antelopes.
But it's nice to envision a fantasy unicorn.