The dentist will make a form fitting tray to hold a teeth whitening gel, which closely fits your teeth.  The gel used by the dentist has been tested for its safety.  The over the counter gels may or may not have been tested.  The reason for this is; there is no government requirement for such testing, as this gel is not considered a drug.  These bleaching gels are usually carbamide peroxide of various strengths of peroxide. 

What will happens when stained teeth are whitened?   The staining comes from eating, drinking tea, coffee or colas.  The stain can also be the result of taking tetracycline antibiotics.  The whitening is the result of actually bleaching the inner tooth, not just the surface. 

Which one of the above mentioned methods is the best?  It depends on what you can afford to pay and the time you are willing to commit to the whitening method. It also depends on the strength of the whitening gel.  The Internet has many websites, which will help you decide.  Enter teeth whitening in a good search engine.  Then read the sites that show up to make an informed decision.