Wiggle the chair back to see where it's come loose. You will see it moving back and forth in the joint where it's come unglued. Look for any broken parts or pieces while moving it around.
Spread a small amount of wood glue on to a thin sliver of wood. Firmly press the sliver into the gap at the joint where the chair back is loose. Place one in the front and back of the joint. These will act as wedges to keep the back from wiggling anymore. Also, because they wedge the back in place, you won't need a clamp to hold everything in place.
Clean up any excess glue with a damp cloth. After an hour, trim the ends of the slivers flush with the joint that you slipped them in to. Use a sharp utility knife for this because the blade is very thin and can reach into tight spaces.