I'm living in an old apartment, circa 1945. The walls seem to be made of an incredibly hard stone or concrete. Any recommendations on the right drill bit type to use? My standard set doesn't even make a dent.
If it is an interior wall, it is made from plaster on top of wood slats. For stone drilling, get what is called a tap-con drill bit. You are also going to want to use a hammer drill to make your holes. If it is stone, this bit will also work on stone or concrete like an exterior wall. Be sure to use a hammer drill.
For drilling stone or concrete you'll want to use a good quality masonry bit, and a good rotary hammer drill.
As Shirlock points out quality is important here, so you don't want to skimp on the tools when drilling stone or concrete. You'll want to look for a carbide tipped masonry bit, as these tend to last a bit longer than the typical masonry bit. You'll also want to make sure the bit is rated to be used in a hammer drill, since a standard bit can easily be destroyed by the hammer action of the drill (causing personal/property damage and/or death).
If you're doing a small job and don't want to spend the money on expensive tools, you should be able to rent what you need from a local home center (Home Depot, Lowes).
A hammer drill is necessary when drilling stone or concrete, since grinding away the material with a standard drill would take far too long. A hammer drill works similar to a normal drill, but with an added mechanism that "hammers" the bit while also spinning it. The idea is that the hammer action will break up material at the bits tip, then the bit will grind up the loose material and carry it away along it's threads (clearing the hole).
To reduce heat on the bit you'll want to drill at a slower speed, and pull the bit out of the hole from time to time to clear the debris. Remember the hammer action can be brutal on the bit, so you'll want to use a quality bit and keep it cool to reduce the possibility of shattering it.