"La Suisse a beaucoup de jolis villages."
Translation:Switzerland has a lot of lovely villages.
La Suisse is the subject. It is singular, and so we use the third person singular conjugation "a."
Villages and village sound the same, you are right, but there is another clue that tells you that villages must be plural. La Suisse a beaucoup de/ Switzerland has a lot of, a lot of things has to be plural. To have a lot of things you must have more than one, hence villages.
Don't ask what delineates between village and town or city. Most people don't know. But, I will add one interesting thing. There used to be small villages apart from anything but perhaps surrounding farm land. Today in America many small sections of a city are called villages. I live in one. Even i this small city (about 40,000 inhabitants) there is (at least) this one village which is barely anything resembling an old-time village. At one time it actually was a small village apart from anything, but the city grew and captured it.