What is the best kind of artificial valve?

It very much depends on the situation.

The need and ability to maintain long-term anticoagulation (blood thinning) is a very important factor. Several classes of people may not be good candidates to take warfarin (brand name coumadin), and therefore should receive a bioprosthetic valve. This group includes young women who wish to become pregnant. They should not take warfarin since it can frequently cause birth defects. Some people wish to continue to lead very, very active lives and taking a blood thinner would complicate the ability to do so. People who have had previous and sometimes repeated problems with bleeding (for example, frequent bleeding from ulcers) are often felt to be better served with the bioprosthetic valves which do not require anticoagulation.
On the other hand, for people who are going to need to have anticoagulation anyhow because they have the abnormal rhythm known as atrial fibrillation, using a bioprosthetic valve would have little advantage since they are going to need to take the blood thinner anyhow. They will most often receive a mechanical valve to take advantage of its longer life.
Another consideration for some people is the efficiency of the valve. That is, the replacement valve is never as good as the "real thing" in terms of how well it opens. Some artificial valves obstruct the flow of blood quite a bit, while others (mainly some of the bioprosthetic varieties) obstruct very little. There are some world-class athletes who compete despite having artificial valves, and they will have one of these "low profile" valves utilized to avoid loss of efficiency. This is generally not a consideration for most of us.