The ebvious answer is a four legged, furry canine.
In terms of vises and benches, it refers to a means that work is secured to the benchtop. In the bench top it self, there are a number of metal (or wood) objects that slide in and out of holes in the bench top to hold on end of the stock. These holes can be square or round. Lee Valley has a premium line of bench dogs.
All of these holes and dogs line up to the vise. The vise has a small metal "dog" that slides up to catch the other end of the stock. The vise is tightened until the stock is secure. The reason that it slides down is so that stock can be slid over the top to the vise without catching.
I have a cheapy (made of) wood vise that has holes drilled in it. I slide the dogs into them and holes in the bench to secure my stock. I cut off some 3/8" bolts to make my bench dogs.