A graphics card capable of rendering OpenGL. A Riva TNT2, released in 1999 or 2000, should do. Stellarium is also fairly processor intensive, so you will get higher framerates with faster processors. Any reasonably recent computer should be able to run Stellarium. Furthermore your computer should be running a Linux, Windows or MacOSX operating system.
To give you an idea of what's possible, we've had success with these older computers:
Pentium III 800Mhz, 128MB RAM, Intel on board graphics (under Windows XP).
AMD K6 400 Mhz, 256MB RAM, Nvidia Gforce2 AGP, DirectX 9.0 (Windows XP)
Mac PPC's back to about 667MHz, running at least OSX 10.3.9, any Mac Intel
