If you're just using the machine for basic needs (more or less anything except video processing or modern gaming), then you'll be fine with integrated.
For games or video processing, a dedicated card is a must.
If you choose integrated, it'll share its memory with your RAM, so for example your 512MB may become 480MB to the rest of the system. Make sure it has enough RAM (1GB is a must really, for Windows).
missed the accelerator (I smell a homework question)
int and standard are nailed by plow and fo3
the accelerator card is (was - not seen one for a few years now) an expansion ~ (generally extra vram) to improve the performance of (generally) the integrared option (although the matrox millennium card offered accelerator cards as well) .... in the days when memory cost more than the pc