Roller covers are made from a range of materials that have different 'nap' lengths; the type and nap size you need depends on your project. Choose a roller cover that won't leave lint or roller marks behind; don't even bother with the foam rollers--they're fine for testing paint colors, but on large surfaces they just create thousands of tiny air bubbles in the paint film that leave behind a 'cratered' finish.
Good roller covers have moisture-resistant plastic or phenolic cores that won't absorb water and lose their shape. They're also made with solvent-resistant glues, so that the fabric nap will stay bonded to the core instead of ending up on the wall (shed resistance should be listed on the label of any good sleeve). Avoid roller covers with untreated cardboard cores, the kind sold in those bulk packs, they will absorb paint, get soft, and lose their shape in no time.
If you're painting smooth new drywall or slab doors with high sheen paint, choose a roller cover with a short 1/4-inch or 3/16-inch nap. A short nap roller cover won't hold a lot of paint, but it will leave a very smooth finish with very little stipple, or texturing. For matte and eggshell paints on smooth, previously painted drywall, use 3/8-inch or 5/16-inch nap sleeves. The slightly longer nap will hold more paint with minimal roller stipple. A quality roller cover with a 3/8-inch nap is versatile enough to work with most paints on most surfaces.
For medium-textured plaster and similar surfaces, use 1/2-inch nap sleeves, but keep in mind that roller stipple will be noticeable. This is a good choice for previously-painted walls or ceilings that already have some texture to them. For rough surfaces like concrete block, most pros use 3/4-inch or even longer nap sleeves.
The denser the pile, the more paint the roller cover can hold without splattering. For best results, you're looking for a roller cover that will hold as much paint as possible, yet release it onto the wall with minimal splattering (these are often called 'High Capacity' rollers). Lambswool or sheepskin roller covers are dense and extra-long, so they hold a lot of paint. They're good for textured walls, but won't leave a smooth finish on a smooth wall. For the smoothest finish on the smoothest walls, choose a micro-fiber or mohair cover. "Mohair" covers are also good for applying smooth finishes such as polyurethanes or varnishes.
Finally, get an extension pole along with your quality roller covers. The extension pole will allow you to roll from the ceiling to the floor applying a good even coat of paint.
Good frames and roller covers are worth saving, so they should be cleaned carefully after use with soap and water. With today's fast-drying paints, it's also good to know that roller covers can also be temporarily stored between coats without cleaning: Simply wrap the wet cover in a plastic bag, seal it up in another plastic bag, and store in a cool, dark space. Kept this way, the paint-filled roller cover will be good as new for several hours and even overnight.
