Chairs are like people. When they get old their joints start to give them trouble. They loosen up, begin to squeak and eventually falter. But with proper care, this deterioration can be arrested and the chair can serve a useful life for many more years.
Prompt treatment is the secret. If you treat a loose joint soon enough, the cure is easy and reduces or even prevents further trouble.
At the first sign of looseness, treat the joint with a preparation such as Chair-Loc, a liquid found at any hardware store. It seaks into the loose joint and expands the fibers of the wood, tightening the joint.
If you ignore a loose joint for long, it will wear against itself and become even looser. The wood fibers will become crushed and irelastic, and won't respond to a simple Chair-Loc retreament.
Regluing is the only cure. If possible, disassemble the joint. Clean off all the old glue with warm water. Clean both the mortise and the tenon. Reassemble the joint with glue and check the fit. If the parts fit fairly snugly, let them dry thoroughly (a day or so) then reglue with epoxy. Epoxy glue does a good job of filling the gaps in a loose joint. (Note: Don't use fast-setting epoxy; these quick-setting glues are much more brittle than the epoxies that take a day to set.)
If the joint is still very loose and sloppy, after you put it back together, simple gluing won't work: You'll have to tighten the fit, too. Use wedges whenever possible.
When you reglue, make a simple tourniquet clamp to hold the joint in place while the glue sets. Or use one of the web clamps sold at hardware stores.
Often you can't disassemble a problem joint without taking apart other perfectly sound joints. This makes effective repair difficult; you won't be able to clean off the old glue, or coat the parts thoroughly with fresh glue. And adding wedges will be impossible unless the tenon goes all the way through its mating part.
A glue injector can help you get glue deep into the joint, or you can try to work glue into the joint with a thin silver of wood and by wiggling the loose parts. In any case, use white glue, not epoxy. White glue will probably be more compatible with the residue of the old glue.
If the end of the tenon is accessible you'll be able to wedge it. Split the end of the tenon with a chisel, then gently drive in a wedge cut from hardwood (a clothespin is a good source). Coat the wedge with glue before driving it into place. When the glue is dry, trim the wedge flush with a knife. If you can't use a wedge, turn to the toothpick shim technique.