I have a door with an old-time door knob, pictured below. What is this style of knob hardware called? What are my options for replacing it?
It's a Mortise Lock.
So named because of the rectangular mortise in the edge of the door the mechanism fits in, as contrasted with "modern locks" that mostly use intersecting round holes.
You will have a job getting the mechanism out of the hole with ~27 layers of paint over it and its screws. Hopefully it's obvious that you'll need to remove one knob and the shaft (still connected to the other knob) to slide the mechanism out the edge of the door.
Specialist hardware suppliers will have some options. Other hardware stores can probably order one, but may not stock any, unless the housing stock in your area provides enough business in replacing old locks in the old style for them to do so.
There are many options for repair or replacement of mortise locks. Parts like knobs in all kinds of materials are available from antique dealers or new. Internal parts like springs are available from locksmiths. If you want to replace the whole thing with something more modern that fits in the existing door, you can buy simple modern mortise locks. If you want to replace it with an ornate but new replica, those are available from specialist historical hardware web sites. If you want to replace it with a not-broken identical one, those are available from antique dealers.
While you certainly can remove/replace that doorknob as others have suggested, IMO the "correct" way to replace that is to replace the whole door. You can buy a pre-hung door from any of the box stores for around $150, a door casing kit for another ~$30. It's not very difficult to replace a door, and certainly easier than trying to bondo over the hole left from that knob.