St. Andrews is a bit like Mecca for golfers. Here is where it all started, less than a mile from my hall at the Old Course. Everyone who is anyone has played here. Tiger Woods won the Open last August, but more famous (shocking, right? More famous than Tiger?) players have included Mary Queen of Scots, King James IV (only after the ban on golf was lifted), Tom Morris and his son Tommy, and more recently, Prince William (a graduate of St. Andrews himself). Just about everything along the northern part of town is devoted to golf, from the street names:

The Links at Golf Place
to the shops:

The brand-new Old Course Shop
along with numerous hotels and bars. Incredibly exclusive golf clubs are abound, including the Royal and Ancient Golf Club.
The golf courses themselves are beautiful, and most of them overlook the ocean:

The Old Course
The golf clubs themselves are constantly looking out for the quality of the course. The R&A even has their own recommendations and guidelines for what makes a good course. I think there are six courses within St. Andrews, and Course No. 7 is on its way.
On a more personal, opinionated note, Course No. 7 will be the most beautiful course probably in the world, although I have no basis for this other than that I run by the construction site every day. Using the link above you can see from the pictures that the course sits atop a cliff on the North Sea, and this time of year when the flowers are out and the ocean is an incredible blue there is no sight more amazing. As long, I'm sure, as you don't hit your ball into the sea.
(For more info on golf in St. Andrews the St. Andrews Links Trust is a great resource for history and tourism.)