Is there a piece of wood that goes on top that's not in the picture?

There are ways to fix this but only attempt them if you have no intention whatsoever to bring the chair to a professional. The hardest part about repairing chairs is undoing the damage caused by previous, unsuccessful repair attempts. It takes a long time to clean glue residue out of the wood and nothing will fit back together until that's done.

I'll list three approaches that are all viable but need to be done right to prevent permanent damage. With that disclaimer out of the way...

You can use a saw to cut the the tops of the legs off about a half inch above the top rail. Sand a nice bevel or round-over on the top edges of the legs and smooth the saw marks with sandpaper as well. Obtain a few cans of stain in about the right color and use the off-cuts from the leg to test color. Mix small portions of stain in a clean container until the color matches the original chair. Stain the tops of the legs then topcoat with polyurethane or lacquer of similar gloss to the original chair.

Option two is to clean off all the old glue from the joints and use a slow set epoxy to reattach the parts together. Slow set means the epoxy will have an open time of 15 minutes or more. Fast set epoxy is not good for joinery and has little strength or durability. The downside to using epoxy is that trying to purchase the good stuff without buying a large quantity could be difficult. It's also messy and will get all over the chair so everything that needs to stay clean has to be waxed or wrapped in plastic. Masking tape could stick to the chair and damage the finish if epoxy soaks into it.

Option three is to remove all the old glue and old dowels or tenons and drill new dowels to reattach the parts. If you aren't a seasoned woodworker this is probably an impossible task so if you don't already know how to do it, stick to one of the above two approaches.

What is that yellow piece in the corner?