Most of us use a regular domestic machine to do both quilting and piecing. To quilt on a regular machine, you still have to baste the quilt. Spray basting often works well for this. To quilt the middle, half of the quilt has to be under the arm; I loosely accordion pleat to fit that amount of quilt under the arm. (Alternatively, you can cut the batting into 3 pieces to reduce bulk under the arm.)
With a regular domestic machine, there are two basic ways to do the quilting -- using a walking foot, or free-motion quilting using a darning foot. A walking foot is great for straight lines or gently waving lines. For squiggles or patterns, most people drop the feed dogs and use a darning foot. In both cases, you are moving the quilt to quilt.
The other basic way to do quilting is to buy a machine and frame setup. You need to have space to set up the frame. The machine rides on a platform above the frame. Instead of basting the quilt, you mount the layers on the frame. Instead of moving the quilt sandwich around while quilting, you move the sewing machine. These machines have a different needle orientation than regular sewing machines. Basically you are facing the needle-end of the machine, and the machine has handles on each side so you can move the needle around.
Most people start out quilting on a domestic machine and only graduate to the more expensive frame setup once they are sure it is worth the space and investment for them. If you buy a domestic machine, just make sure you buy a standard size. 3/4 machines (I think the Janome Jems fit this category) can be great for piecing and carrying to class, but the space under the arm is very tight for quilting. A few domestic machines have a larger area under the arm (I think the Janome 6600 is an example).
Here is a Youtube video that demonstrates quilting with a domestic machine:
http://www.youtube.com/watch?v=LHv2GV8F-8E
There are many other Youtube videos that demonstrate quilting both on domestic machines and on frame setups.