You'll want to use a really high-quality paint designed to withstand the wear and tear common to doors. You'll also want to use a paint that will level well and minimize brush strokes.
You do not have to remove the door in order to paint it properly, but you should use painter tape to mask off hinges, the knob and other hardware. Remove any loose or peeling paint, fill any dents or holes with with wood filler or spackling compound and sand the door smooth and dull with sandpaper (150-grit works well for this purpose). Remove any dust, debris or dirt with a damp rag or tack cloth.
Prime the door with a high-quality acrylic primer and let it dry. A combination of paint brush and roller works best for painting doors. Use a brush for painting the edges and around the hardware, and fill in the rest of the door with the paint roller.
It is always a good idea to use a paint that is specifically designed for the specific type of surface you are going to paint. This means that if you are going to paint a door, get a paint that is recommended for painting windows and doors. The Benjamin Moore Aura in the Satin or Semi-Gloss finish is best for this use.
When you are looking for paint colors it is a good idea to use whatever you are using for the trim color. You can either go with the color that was already there or choose something else that would coordinate with the color such as a lighter or darker shade of the same color tone.
Start with the roller to get the paint on the door quickly and evenly. You will then go back with the brush to paint the edges. A trick for removing any orange-peel texture left by the roller is to lightly drag the paint brush over the rolled paint immediately after rolling. Make sure it is a high-quality roller cover and brush to achive the smoothest finish.
Another thing you can do to achieve an even smoother finish is the add a paint conditioner to the paint. This paint additive is a best-kept-secret of the best painters and levels the paint to a super smooth finish. Floetrol works well for regular latex paints and the Aura Paint Extender should be used with Aura paints. You simply mix in about 8oz to a gallon of paint before painting.
If you are working with a paneled door you will want to start at the top and then go to the bottom. You will start with the roller again and then brush off excess paint and smooth out the surface. You will also want to have a rag because this type of door is prone to dripping and you will want to wipe it off as you go.
Once you have painted the door let it dry completely. After this first coat sand it gently, and then apply a second coat. Once this second coat is dry you can put the door knob and other fixtures back on and stand back and admire the job.