The center of the tooth contains a chamber that contains the tooth's main nerve as well as blood and lymph supply. When that nerve space is injured (usually from decay or a fracture) it cannot repair itself. It becomes inflamed, and then dies. Later the pulp area with its dead tissue can become inhabited by bacteria causing an infection or dental abscess.
A root canal (or endodontic therapy) aims to remove the dead tissue from the pulp and replace it with clean filling material. Once there is no longer dead material in the pulp, and no longer an empty space, any bacteria that were living from it will be evicted and die off.

There are also times when a tooth needs to have a root canal in order to allow it to support a crown or other prosthetic dental work. When a tooth is broken or substantial tooth structure is missing, your dentist may need to do a root canal so a post may be placed inside the pulp area allowing a crown to be properly attached.

A tooth which has had a root canal may already be weak due to lack of tooth structure. Also, a tooth becomes a bit more brittle after having a root canal. Your dentist will be able to tell if the tooth needs a crown to protect it from breaking later. Some teeth should always be crowned after a root canal. These are multi-rooted teeth that are especially to fractures which split the entire tooth apart.

Root canals allow patients to keep their own tooth, still happily living in the mouth but with no infection. No replacement is as good as a person's own natural tooth. None, period.

Root canals do not hurt because the patient is given local anesthetic. The days of root canal torture are long over. If a tooth is infected, an antibiotic may also be given. This may be amoxicillin, or in case of penicillin allergies, erythromycin. A few dental infections may need clindamycin if they are infected with a bacteria that is not susceptible to amoxicillin. If your tooth is not dead but not infected, you will not need antibiotic coverage in most cases.

Some teeth have only one root and others have as many as five. Every root has a pulp canal that must be located and cleaned. You can see that some teeth are more difficult than others to treat. Sometimes the canals are calcified and this can be another problem the dentist must solve. Most root canals today are successful, but there are some that are not. This may be because the tooth is involved with periodontal disease or because there is a small hairline crack in the tooth that cannot be detected. Sometimes when a root canal is not successful, a surgery can be preformed that exposes the root and the tooth is treated at it's root end. This is usually done by an endodontist or oral surgeon and it is called apical surgery.
Root Canal saves many teeth that would otherwise have to be extracted. It is safe and patients are comfortable during the procedure. There are a few anti-root canal theories but none really makes sense. The Internet is filled with strange theories that when checked do not stand up to scrutiny. (I have seven root canals in my own mouth and I have had them all for some time, allowing me to keep natural teeth, avoid implant surgery, or tooth loss or very expensive bridges.)

The drawings below will explain how a root canal is preformed:

The first drawing shows a periapical tooth abscess. Here, decay has entered the pulp chamber (chamber in the tooth's center where the main nerves and other tissues live).  This infects the nerve and blood supply, which becomes inflamed and eventually dies. Bacteria take up residence in the pulp chamber and canals and live on the dead tissue.  The infection produces toxins, blood, gases, etc. and these impinge on the area around the root of the tooth, pushing on the surrounding bone.  The infection looks like a little pocket around the end of the tooth on an x-ray.  This infected area can be very large or quite small.
Decay is not the only thing that can kill a tooth.  Other causes include cracks in the tooth, trauma like falls and auto accidents (Wear a seatbelt) and sometimes, orthodontic treatment.

The second drawing shows how the tooth is cleaned out.  First an access  hole is made in the top of the tooth. The cleaning is done using long files.  These files come in different widths and a progression of sizes will be used, each one cleaning the sides of the canal a little more. The tooth will be medicated to kill any bacteria in the pulp.

In the last drawing, we see the finished root canal: the tooth  pulp has been cleaned out.   The empty canal space has been filled with cement and plastic material, so that bacteria are evicted.  Note that the area in the bone where the pus  pocket formed is healing with new bone. A filling has been placed in the top of the tooth. In many cases, the tooth should be crowned to keep it strong.