Links-style describes those treeless courses you see in Britain. I believe it is the oldest style and is based on the natural terrain found along the coastlines. They were developed at a time without golf architects and earth-moving equipment, so they reflect a kind of minimalist design. They have few trees, tall grass rough, pot bunkers, and rolling fairways. I think every British Open is played at this type of course. They are very frustrating to play.
Yes, well, as with almost everything else, the American versions of what other countries do turns out better IMO, including beautiful tree-lined golf courses.