You buy sandpaper in two grits, medium and fine, if it's a wood dresser with a clear finish over the bare wood. If it's painted, buy coarse, too. I recommend a sanding block and sanding sponges to supplement.
Now you get your dresser in a well-ventilated area and sand, starting with the coarsest grade wrapped around your sanding block. Remove drawers and take off the pulls, putting them and the screws in a plastic bag. Sand until you see bare wood all over it. (Safety: Most people wear a breathing mask [and eye protection] to keep from breathing floating particulate matter they sand off.) Minimize transfer of sanding dust by cleaning up after each session of sanding.
Repeat with the finer grade, and the finest grade, until you have a smooth surface. Wipe it clean with a damp rag you rinse often. Let it dry completely, then paint using an enamel and a good brush. Expect to do at least two coats for complete coverage.