When you're disgusted by something, or simply dislike it, you have a distaste for it. You might have such a distaste for the smell of meat cooking that you only go to vegetarian restaurants.
Some people have a distaste for living in the suburbs, while others feel distaste at the thought of living packed into a large city with millions of other people. You might have a distaste for the sport of boxing, while your boxing-loving cousin's distaste is for ice dancing and synchronized swimming. When something's just not your taste — you don't care for it — you can say you have a distaste for it.