FSO will run reasonably well on any card that suports at least OpenGL 2.0. For all the shadery goodness (including normal maps), OpenGL 3.0 is needed.
Alternate version: If your card supports Shader Model 3 (which any ATi or nVidia card currently on the market does, IIRC), you can get the full enjoyment.
In terms of Memory, 1+ GB should be enough, as always though, more can't hurt.
In terms of CPU, you need to look at single-core performance, since FSO only utilizes one core at any given time. In other words, this is a game where GHz still matter greatly.
That being said, I _think_ a late-model P4 should be able to run it smoothly with all enhancements.
With newer builds, using the SSE or SSE2 variants (which include a few processor-specific optimizations) will yield a marginal to noticeable performance boost. In addition, the various improvements made in 3.6.12 result in a very noticeable performance boost across the board there as well.