If you write post-apocalyptic fiction, at some point your characters may need to do some improvised dentistry. Some dentistry, like teaching people how to care for their teeth, is easy and requires little equipment (even toothbrushes can be improvised without too much trouble (piece of rag on a stick), fluoride is plentiful enough in tea and seafood, and toothpaste isn't really needed if you're okay spending a little longer brushing).
Other dentistry, such as using a dental drill or performing a root canal, requires an art and science that isn't easy to improvise or teach quickly to people who aren't dentists.
There is, however, one major dental procedure that can be performed under very austere settings with minimally trained people with improvised equipment: Tooth extractions.
The following post is about how to do that.
A tooth extraction needs to be done when there is an abscess at the root of the tooth, and a root canal cannot be done to safely drain it. It may also need to be done if the tooth is badly broken with an exposed nerve. A main sign that a tooth may need removed is that the tooth is very painful, and the pain is constant. More minor cavities may cause pain, but usually the pain is intermittent or only under certain circumstances, like food in the cavity.
Look for a "gum bubble": swelling at the base of a tooth below the gum line. This is the abscess your characters are trying to drain by removing the tooth.
The first thing that needs done is to reduce the swelling. Swelling means anesthetic generally won't be able to get to the nerve, and won't be able to numb it. Ideally, this is done with antibiotics. Generally, penicillin still works for this. If your characters don't have antibiotics, it can be done less reliably by holding the hottest water the person can stand in the mouth next to the gum bubble. This needs to be done nearly continuously for several days (breaks for sleep ok), until the swelling goes down.
The next thing to do is situate the person so it is easy for your characters to remove the tooth. This picture does a good job showing how to do that, with a straight-backed chair with a cloth headpiece, situating the person on cushions or the operator on a box to adjust height.
The next thing to do is provide anesthetic. This really can't be easily improvised. Your characters will need at the very least lidocaine, preferably mixed with epinephrine. The lidocaine deadens the nerve, while the epinephrine prevents it from escaping the area and makes it last longer.
To anesthetize a lower tooth, inject in the back of the mouth at the curve of the jaw on the side of the tooth to be extracted. This will block (deaden) the nerve that gives feeling to all the teeth on that side of the mouth, and part of the tongue. To be safe, also inject anesthetic at the base of the tooth into the gum. It will take about 5 minutes for the area to become numb.
To anesthetize an upper tooth is a little easier. Just inject into the area above the tooth to be extracted. The bone is spongy and will absorb the anesthetic and carry it to the nerve to be deadened. It will also take about 5 minutes to work.
Once the anesthetic is working, your characters can remove the tooth. It is important to note that fiction has done you wrong here. Just trying to yank out teeth with pliers will break the tooth and cause more problems.
The first thing to do is to separate the gum and make the tooth loose. This is done by inserting thin, flat piece of metal into the area between the tooth and gum, sliding it down until resistance is felt, and twisting until the tooth starts to move. This can be done with a tool called an elevator, but it can also be done with a small screwdriver or an awl that has been dulled and flattened. The goal here is to gently break the attachment that is holding the tooth in place. This will loosen the tooth and prevent the gum from tearing when the tooth is removed. Your characters need to do this to both sides of the tooth.
Then, further loosen the tooth by rocking it back and forth or in a figure 8 pattern with pliers. Ideally, these are dental pliers, as regular pliers (even wrapped in gauze) are likely to break the tooth. But you know what your characters have. Even if the tooth breaks, the abscess will drain, and while a retained piece of tooth does increase risk of infection, many retained root fragments will work their own way out eventually.
Once the tooth is very loose, your character can fully remove the tooth by rocking it (molar) or twisting it (front teeth) until it comes out. It is important to know if fragments are retained. Mention that your character is inspecting the tooth when it comes out to ensure the whole thing was removed.
Then the hole left will bleed. To stop the bleeding, have your character apply direct pressure by squeezing the sides of the socket for 1-2 minutes, then have the patient character bite down on a piece of rolled gauze or cloth for a half hour or so. They should not rinse their mouth for 24 hours. After that, have them rinse with warm salt water frequently for the next few days. Keep the mouth clean and avoid hot liquids.
Pain control can be achieved however your characters usually do that in their post apocalyptic setting.
Information gathered from Where There Is No Dentist by Murray Dickson and Improvised Medicine by Kenneth V. Iserson.