What is the strip that runs between the garage and the driveway called? Also what does it do? Is it a thermal break or is it there too prevent the concreate from cracking? What is the material? How deep does it need to be when installed?
That gap is to relieve stresses due to expansion/contraction caused by temperature changes. It's not a thermal break. You need some kind of flexible filler to keep water from getting into the gap, freezing, and cracking the concrete.
There are numerous materials than can be used to fill in the gap, so it's hard to say what may have been used in your case. Looks like it may some sort of tar/oil impregnated material. It's usually the depth of the concrete pour, probably 4"-6".