It helps if your Sim is an animal lover and/or equestrian. Then they can befriend the unicorn much more easily. I know Carl has a great guide as to how to find them, but I'll summarize. A unicorn can be found between 8pm to 5am. They have a 40% chance of appearing each night and I normally find them in the fishing areas and the waterfalls. I don't know for sure, but my guess is they have a higher chance of appearing in Appaloosa Plains, as that is the 'pets' town.
You'll know a unicorn is there when you see this rainbow, aurora cloud thing between 8pm-5am. Underneath the cloud, a black or white unicorn will be there. I've heard that black unicorns are 'evil' and white unicorns are 'good', but I don't really know. Your Sim should spend as much time as possible watching or interacting with the unicorn, but there is a small chance you can adopt the unicorn on the first night. You just have to keep checking back each night for the unicorn.
If your Horse has the Friend of the Herd Lifetime Reward, Wild Horses will sometimes visit your lot and Unicorns can be among them.
Hope this helped!
Happy Simulating,
Foxx