Hi Colton:
The short answer is no, you don't have to get a new processor if you get a new motherboard. You just remove the current processor, clean it off with Arctic Silver cleaner, reapply Arctic Silver or other thermal compound to the processor, attach the heatsink/fan, and then put your graphics/sound cards in the new board. This assumes that the motherboard you buy is compatible with the processor on your current machine.
The longer answer is that with an emachine, it's likely that you won't be able to replace your motherboard. emachines, like many OEMs, use proprietary equipment. If you find your current machine feels slow, and if it's a couple years old, you may find it more advantageous to save up for a new build that will allow you a better upgrade path.
It may be cheaper--at least in the short term--to add more memory and pop in a better processor if you want added performance. If you're interested in gaming, you may find it more advantageous to see if you can upgrade your video card.
However, if you list your specs, it will be easier to see what might help you in your situation.
--ceolstan