Sound travels through doors in two ways. As a solid barrier, the door itself reflects some sound waves while transmitting others through to the room on the other side. The thinner and lighter the door, the more readily the sound passes through. The other avenue -- and typically the biggest problem -- is through air gaps around the door and frame. The first step to soundproofing is to seal the air gaps. If that doesn't do enough, consider modifying or replacing the door.
Soundproofing a door is much like weatherizing one -- any cracks or crevices that allow air to pass will let sound do the same. To close the gap between the door frame and the wall frame, remove the door frame and apply a soundproofing rubber around the door. Hold the rubber in place and fill in any cracks with acoustic caulk and then reinstall the molding over the rubber.
There is also a gap between the closed door and the door frame, running across the top of the door (the header) and down both sides (the jambs). Close this gap by installing a door gasket designed for this purpose. If you're working on a tight budget, weatherstripping will also do the job.
A door gasket will seal the area around most of the door, but it will not fill in the gap between the door and the floor. For this, you need either a rubber door sweep or an automatic door bottom. A traditional door sweep is simply a bar with a rubber piece underneath it that mounts on the door. The sweep should just barely touch the floor since the door will be difficult to open and close if the rubber sweep is too tight against the floor. Simply screw the sweep to the bottom of the door and it's ready to use.
An alternative is to install an automatic door bottom. This option is a bit trickier to install but is a superior soundproofing option. These door bottoms automatically lift up when the door is opened and then drop down to create a tight seal against the floor when it is closed. You may wish to consider hiring a professional to install this spring-loaded mechanism unless you're an experienced do-it-yourselfer.
Beef Up the Door
Interior doors are often hollow, making it easy for sound to pass through them. For superior soundproofing, replace interior doors with solid exterior doors that are much better at blocking sound. Even if you rent, hanging a new door may be an option so long as you put the old door back up when you move out. Avoid doors with recessed decorative panels, as these make parts of the door thinner.
If you own your home and prefer to work with the door you already have, you can do so by reinforcing the door. Add a 1/2-inch-thick sheet of medium density fiberboard to your door using wood screws. For extra sound dampening, glue the door and MDF together using a special anti-vibration glue. Simply cut the MDF to fit the door and drill a hole where you need the doorknob to be.