Stain soaks into wood fibers to permanently change the color of wood. Sanding slightly scratches and smooths the wood fibers for an even stain. If there are compressed fibers or any glue or sealer on the surface, the stain will not soak in evenly, resulting in light spots.
Some wood has uneven grain and won't stain evenly. Wood with tight, figured grain, such as curly maple, has alternating swirls of tight and open grain that take stain differently. Likewise with the closed-grain ray flakes in quarter-sawn oak. Some cheaper woods, such as plywood, are also prone to uneven staining.
If you have just a few spotty areas, you can spot-treat by sanding the area and spot-staining. Before staining, wipe with mineral spirits to remove any wax or oil that may be on the surface. A surface riddled with spots will have to be sanded to bare wood and primed with a sanding sealer, which helps stain absorb evenly. Alternately, you can use half-strength shellac as a sanding sealer.
Use an all-in-one stain and polyurethane sealer, which both soaks into the wood and sits on top of it. Many coats of the product put more color on top and even out the differences in color. Consider using a light stain, which is less likely to stain unevenly than a darker one. If all else fails, paint the dresser. Your excellent sanding job and one coat of primer will yield a superbly smooth, painted finish.