The work starts by creating what is practically a hydroponic system for growing the grass. When constructing the green, a bulldozer creates a 12-inch to 16-inch (30 to 40 cm) deep hole the size of the green. In the most advanced systems, this hole is completely lined with plastic, and then gravel, drainage pipes and sand are added. The green's grass grows in a sterile sand medium with perfect drainage! The surface is contoured to allow perfect run-off as well, so there is no puddling when it rains.
Where you put the green is also important. It needs plenty of sunlight (preferably full sunlight with no surrounding trees) and good airflow over the green.
Then you choose perfect grass. Several of the links below spend a good bit of time discussing different species and types of grass.
A sterile sand medium and a good location controls for a huge number of variables, but now the grass is totally dependent on its keepers for life support. That means the grass needs a steady diet of water and nutrients to keep it alive. To this mix is added a variety of herbicides (to kill weeds that try to move in), pesticides (to control insect damage) and fungicides (to control disease) to help keep the grass perfect.
Once the green is established, you start in on maintenance. This includes daily mowing with a precision green mower, watering, fertilizing, applying chemicals, aerating, and general coddling.
If you were willing to do all of this, you too could have a golf green in your front yard -- many of the links below will show you just how to do that if you are interested!