There are a number of types of dog around. They mount into dog holes in the bench top in line with the dog(s) in the vice, spaced at maximum intervals of about 80% of the vice jaw travel accross or along the bench, depending on the vices installed and dog capable.
Rectangular dogs can be metal or hardwood,
this is a steel one. Head is larger than the body, and there is a retaining spring on the side to hold the the dog is the users desired position. Typically use a 1in x 7/8th in hole in the benchtop inclined about 3 degrees toward the dog in the vice. Dog hole is enlarged at the top to allow the head to drop below the benchtop when not in use. This is a traditional form, typically used with benchtops laminated from numerous strips timer side by side. Modern technique is to cut the dog holes into the strips using a table saw or router and template before the strips are glued up, but could be cut into a finished top using morticing techniques.
Round dogs can be metal or timber.
This is a typical brass dog. Typically 3/4 inch diameter and various lengths. Dog holes can be drilled or routed into top at required locations once the top is completed, and can be used in solid timber, ply/MDF, or Torsion box tops.
There are other devices that also use round dog holes quite effectively, including
holdfasts,
surface vices that mount to the bench top by a couple of dog holes and add an extra vice to the bench,
surface clamps which combine a holdfast and the screw half of a convention clamp, and
hold down clamps. In some instances these additional devices can be very usefull for supporting the far end of boards held between vice faces, and so round dog holes are sometimes found drilled horizontally into the face edge of a bench top and the legs.
1/4 steel might be a bit light for dogs, for two reasons. One, it will tend to bend away from the work under moderate loads, and two, it will not provide much contact surface area, so could crush into the edges of the material being clamped.