There are many diferent types of drip edge...here's a few http://www.rainguttershop.com/edging.htm
You probably have standard drip edge. The shingles should overhang the edge of the drip edge by 3/8". This prevents water from wicking up between the shingles and the drip edge which will eventualyy rot the roof deck. Usually this goes un-noticed until the rafter tails, fascia and soffit are also damaged
The gutter doesn't have to be slid up under the drip edge and fascia. As long as it's tight to the fascia water will drip into it.