Rubbing compound's main benefit is its availability; you can get some at any auto repair or auto body shop. The compound has a rough grit that can sand down furniture so that the scratches blend in with the finish. Due to the rough feel of this product, it's possible to go too far and gouge the furniture. There are many products with a lighter touch that make for better finishing.
Like rubbing compound, polishing compound is available at any auto supply store. It's intended for use in a similar way as rubbing compound, yet has a much finer grit. Polishing compound helps file away the damaged area and also maintains the natural luster of your wood. It works well for pieces with a glossy finish.
A little rubbing compound goes a long way. You can do one pass with that and switch to polishing compound so you don't ruin the wood. To apply, dampen a rag and squirt a dime-sized amount of rubbing compound on the cloth. Rub the rag over the furniture, working in circular motions. Allow the material to haze up, then wipe it off with a clean rag. Flakes of wood will follow. Repeat, or switch to the polishing compound, and apply in the same way. You don't need steel wool or sandpaper when using rubbing compound.
Since it's furniture you're fixing, and not your car, you'll find products meant just for the task at your local furniture repair shop. You can get most of the hard work done with steel wool or sandpaper. Look for abrasive pumice powder sold in grades from coarse to fine, or a fine rottenstone powder. The stone particles will help you achieve a smooth, polished finish once you've ground down the wood enough to hide the blemishes.