Thanks, Benita. That clarifies it a little. Here's some "terminology" for you. A chair arm usually has a hole in the bottom for a round, square or rectangular pin on the top of the outside rung to fit into. The hole is called a mortise and the pin is called a tenon. There is usually a mortise and tenon joint for the rear of the arm to fit into the first tall riser or rung on the back.
If either of these mortise and tenons are involved with the split or the whole arm is loose, you want to remove the arm and sand just on the pin and in the hole because the new glue that you are going to use might not stick to the old glue. You just want to sand enough to get the old glue off, not remove any wood.
When you have done that, if necessary, you can glue the arm back together. You will need Carpenters Wood Glue, 2 clamps that open wide enough to go across the arm, some cardboard and a damp sponge. Squirt some glue in the mortises and on the tenons if they are involved and wipe it around with your finger for an even coat. Put the arm in place. Squirt some glue on the broken edges of the arm and wipe that around for an even coat on both edges. Put the pieces together. Clamp one end then the other putting some cardboard under the holding surfaces of the clamps so they don't mar the wood. Tighten the clamps just enough that you get some glue squeezed out but not so much that you squeeze it all out. Wipe off all the squeezed off glue with a damp sponge and then rinse the sponge well. Let this sit for 24 hours. Remove the clamps and you are done.
I still had some questions about exactly how your chair arm was broken since a broken arm usually involves the mortise and tenon joint but this should be enough info for you to do it. If you have any more questions, let us know. Good luck.