Sound travels in waves, and the key to any soundproofing is to absorb or deflect those sound waves. You could spend a lot of money and completely remodel your room, but soundproofing can be inexpensive. Many people with at-home recording studios for music, podcasting and so forth, need a soundproof room to muffle the noise from the rest of the house.
Pick the right room. If you have a choice of which room to soundproof, consider one on the top floor (if you have more than one floor) since it is difficult to soundproof the ceiling. Choose a room farthest away from noise and appliances. If the walls of the room are unfinished, like a garage, add insulation and drywall.
Clear the room. Remove all wall decorations along with any furniture that is not essential. Sound moves best on flat surfaces, so remove any tables, bookshelves, or cabinets. If the room has a closet, decide whether to soundproof the doors or remove them and soundproof the closet walls.
Carpet the floor and ceiling. Thick shag carpet is an ideal sound absorber. Unfortunately, you have to walk on the floor so it will need to remain even. The floor and the ceiling should be the only flat surfaces in your room.
Cover all windows with cardboard (or fibreglass) and insulation. If possible, use fibreglass or drywall over the windows.
Eliminate all flat surfaces and replace with uneven surfaces by applying carpet to the walls in a wave pattern.
Trap more sound waves. If you have the time and resources, cut carpet into equal size squares (6 inches will do), and cut cardboard squares the same size and cover them with carpet. Staple the squares to the wall so they jut out into the room. If the mini carpet walls are placed close enough together, the sound will bounce between the walls and dissipate.
Soundproof the door. Cover the inside of the door with carpet (Leave room for the door knob.), and apply weather stripping until the door fits as snugly as possible into the frame.
Buy remnant carpet, or ask friends and family for their carpet when they replace it. If you have the money, add fibreglass sheeting between the drywall and insulation inside the walls. If you are going for a completely soundless room (as many university physics departments have) the floor would need to be uneven with carpeted pieces sticking up and a mesh walkway suspended above the floor. You can also apply acoustical foam within the walls, but it is expensive.
Soundproofing is not pretty, so warn family or housemates before you attempt it.