Chances are a chair that wiggles, shimmies or rocks is loose, but nothing is broken. This can easily be repaired.
The first consideration is the type of chair. In a typical dinette set (informal), the chairs have legs that are not perpendicular to the floor and all the joints are glue joints. The legs are glued directly into holes in the bottom of the seat with no screws. Dining room chairs (formal) typically have legs parallel to each other (or nearly so), perpendicular to the floor. The cushioned seat is attached with screws, and the corners of the frame immediately below the seat are held together with a block in each corner that is screwed and/or glued in place. Most chairs fit into one or the other of these two categories, or perhaps combine features of both.
Step 1 - Label the Parts
First put a piece of masking tape on each part of the chair to mark its position. RF=right front, LF=left front, etc. Mark all four legs, the stretchers that run between the legs front to back on each side, and, if there are any, the stretcher(s) running left to right. Mark the stretchers so you can tell which end goes in front, back, left or right. These pieces may look symmetrical but chances are they aren't. They must go back in the same position they were in originally. With a formal chair, also mark the rails, those board-like pieces immediately beneath the seat cushion.
With a formal chair, remove the upholstered seat and the screws holding the wooden corner blocks in place. Number the blocks and the inside of the rail so you can put the blocks back where they came from.
Step 2 - Dismantle the Chair
See what you can pull apart just by wiggling and pulling on the pieces. After you've removed what you can, go after the stretchers, if they haven't already come out. Use the mallet to hit the leg, swinging parallel to the stretcher. Hit as close to the joint as possible, holding the stretcher tightly. Continue this process until the stretchers are removed.
Having removed the stretchers, the legs should be looser than they were, if not falling out. Use the same process to separate the legs from the rails on a formal chair. On an informal chair, turn the piece upside down, striking the seat bottom with the mallet while holding the leg to be removed. Always try to hit as close to the joint as possible, swinging in line with the piece you're trying to remove. You want to pull it out, not break it off. Do this over a padded surface. If the piece separates suddenly, remember you're holding only part of it; the rest will fall.
One last note: some joints will be just as tight as the day they were originally glued. If you can't get a joint apart without extreme exertion, leave it alone.
Step 3 - Remove the Old Glue
Hold the knife at a right angle to the dowel/tenon and scrape the old glue off. Don't cut — scrape. Get it off all the dowels and the tenon ends of the stretchers. Scrape the glue out of the holes that held the dowels and tenons. Again, scrape. You don't want to cut the wood down, just remove the old glue. If the joints are not cleaned properly, the new glue will not adhere.
Step 4 - Dry Fit the Parts
Using your masking tape markers as guides, put the chair back together. Don't glue it yet. This is a dry fit, to make certain you've thoroughly cleaned the holes and not left any burrs elsewhere that will hinder the assembly when you do glue it. Correct anything that doesn't fit.
Step 5 - Glue and Screw It back Together
Whether you're working with formal chairs (cushion seat) or dinette chairs (legs attach directly to the seat) here's the assembly process. Fold the newspaper to get a square four or more layers thick to catch any glue drips.
For dinette chairs: using a Q-tip, spread the glue (you want to get a film of glue — if the glue runs, you've got too much) over the tenons of the stretcher and into the holes the tenons go into. If there is a left to right stretcher, fit it into the two side stretchers first, then insert them into the legs. Spread the glue over the leg tenons and their matching holes in the seat, and insert them. On a chair that was just slightly loose before, you may have to use the mallet to drive them in. You should give them a good tap, anyway, just to make certain you drive them home. Set the chair upright on a flat surface.
For formal chairs: spread the glue as before to attach the front rail to the two front legs. Assemble stretchers as above, then put the side stretchers into the front legs. Put the side rails into the front legs. Lay the chair on its back on the floor. Position the stretchers and side rails over the holes and drive them into place with the mallet. Set the chair upright on a flat surface.
Step 6 - Bind the Chair for Drying
Take a length of sash cord long enough to go around the chair at the feet, and tie a knot in it. The cord should be slightly loose. Insert a section of dowel rod between the cord and the chair, and turn it clockwise to form a tourniquet. Keep turning it to tighten the cord and drive the tenons completely into place. Angle the dowel so it catches on the chair seat (or a stretcher) and can't unwind. Dip a rag some water and wipe off the squeezed out glue. Dry the joints with another rag. Set it aside overnight on a flat surface. This lets you be sure all four legs are touching the ground and you haven't pulled the chair out of line.
For formal chairs, take two sections of sash cord, one around the rails, the other around the legs at the stretchers. Wind up both with dowel rods uniformly to tighten the joints. Wipe off glue as before and leave to dry. Corner blocks can be replaced after the frame has set up.
Be sure to put the chairs on a flat surface while tightening. This insures that all four feet meet the floor.
This guide will help you to avoid unnecessary trips to the furniture store. Repairing your chairs yourself will save you big, and isn't too difficult a task.