how can I fix my dining room chairs?

modern, or old, the joints on chairs as you have,tend to dry out work loose rock from front to back untill the joints eventually part.
go to any d.i.y store and buy some PVA WOOD GLUE and some masking tape, gently tap the back of the chair until the joints part, you will find that you may have also to release the stretchers joints these are the wooden rails that we all tend to rest our feet on at the bottom of the chair, the more joints you can release the better, with the back removed use the masking tape to go arround the joint you are about to glue keeping the glue from the polished edge, sparingly apply the pva glue this stuff is messy so put your chair on an old blanket
when you have put the glue on all the joints gently tap the back of your chair into position as it was, now you need to keep it in position untill dry, to do this after wipeing all pva arround joint so it does not look like a kids ice cream cornet on a hot day,get a piece of strong cord enough to wrap arround the chair and a bit more, get a piece of stick about a metre long entwine the stick with the cord to make clamping motion as you twist it arround wedge the stick in position over night untill the pva goes hard, remove masking tape.
take care not to use to much pva arround joints this stuff is as good as a welding joint when dry,
good luck, take time your chairs will last you a life time,
honestwelsh retired upholsterer

honestwelsh
January 2008