How can I remove a reddish brown stain from an old coat rack so I can stain the wood black?

The thing about stains is that they soak into the wood. Even stripping the old finish is unlikely to remove all existing colour, besides being a lot of work. Since you’re interested in making your coat rack black, then spray painting directly over the existing finish is probably the way to go. A light hand sanding with 180-grit paper will etch the current surface for maximum adhesion of the new paint, and spray application can give you very smooth, brush-stroke-free results. Remove any metal hooks or hardware before you begin, then apply two or three coats of paint. There are many brands of spray paint on the market, and though most will probably work fine, my favourite is called Krylon Fusion. It’s tough enough to stay on plastic, but I also happen to know from experience that it works very well on wood. Besides a superior formulation, Krylon also includes a great spray nozzle that’s much less tiring on your finger than standard nozzles. Krylon does cost a bit more than other spray paints, but when results count, the few extra dollars is worth it.