While I mostly agree with the answer offered by cegaton, in my opinion, it has one characteristic that makes it undesirable in some circumstances. That characteristic is that the center point has a number of duplicate vertices in the center point. In this class of cases, I have found a useful solution in turning the triangle fan into a quadrangle fan.

Starting with cegaton's first illustration, instead of deleting the center vertex, (as at 'A' below), instead deselect everything, switch to edge select mode, and select every other edge between the center vertex and the surrounding edge loop, (as at 'B' below), and delete those edges. Then create quadrangles with the remaining edges, with the result as in 'C' below.

This achieves a similar appearance to the option provided by cegaton, with the exception that it has only a single center vertex.

One warning: this works best with cylinders which have an even number of vertices in the circumference.

While I mostly agree with the answer offered by cegaton, in my opinion, it has one characteristic that makes it undesirable in some circumstances. That characteristic is that the center point has a number of duplicate vertices in the same location. In this class of cases, I have found a useful solution in turning the triangle fan into a quadrangle fan.

Starting with cegaton's first illustration, instead of deleting the center vertex, (as at 'A' below), instead deselect everything, switch to edge select mode, and select every other edge between the center vertex and the surrounding edge loop, (as at 'B' below), and delete those edges. Then create quadrangles with the remaining edges, with the result as in 'C' below.

This achieves a similar appearance to the option provided by cegaton, with the exception that it has only a single center vertex.

One warning: this works best with meshes which have an even number of vertices in the circumference.