Hi Samantha, the best answer to your question depends on how the desk was painted in the first place. If the paint is already peeling... chances are that it was not applied properly in the first place.(i.e. oil based paint applied over latex finish) If you want the most durable finish, my advice would be to strip the desk with citristrip to make sure that there is not additional peeling. Once the old peeling paint has been removed you can start fresh with new paint. I recommend water based paints such as latex - they're much easier to work with. Or you can choose to work with Chalk Paint or Milk Paint which are two types of paint created to refinish furniture. They are ideal because of their ability to "stick" to furniture without sanding or priming.
I could go on forever about this topic, but to keep it short.... you can go to my blog at www.whitecottageboutique.com to find a bunch of tutorials and tips on how to refinish furniture and what products to use.
Hope that helps.
Chemical strip or sand. As with most paint "failures" the problem lies with improper or hasty prep. In practice you should spend twice to three time the effort in the "prep" phase.