It can be done on a "domestic," or usual, sewing machine where the machine is stationary and you move the fabric around freely under the needle as it goes up and down in one place.
This is usually termed "FMQ" or free-motion quilting.
Alternately, it can be done on a "midarm" or "longarm" machine, where, as you say, the quilt is mounted on a frame such that the quilt rides between the needle and the plate of the machine; and the machine rides along on a carriage, while the quilter stands in front or back of the frame and moves the sewing machine around over the quilt.
This is called sending your quilt out to be quilted, unless you have one of these quilting systems!
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