This information is only a general overview. The internet has countless sources. Briefly, the techniques indicated herein are the major elements you are likely to employ. Listed in order of estimated effectiveness.

Staggered double-studded wall The acoustic energy striking one side of a wall is transmitted by the studs to the wallboard on the other side. By using double-stud construction, sound energy is decoupled from the other side of the wall. A prime wall construction technique. Resilient channel Flexible metal strips are screwed perpendicular to the wall studs or ceiling joists. By attaching the wallboard to the flexible strips, the transmission of sound is substantially reduced (lost in the flexing.) This is a must for ceilings and effective for walls. Sound barrier Heavy sheets of vinyl weighing one pound per square foot and loosely attached to wall studs or ceiling joints block (absorb) substantial amounts of energy. Double layers of wallboard Special types of sound suppressing wall board. Acoustic sealant Seal all floor joints, cutouts for AC receptacles and pipes with flexible sealant to stop sound from leaking through into other areas.

Other Considerations

1. Air Handling Ducts
   Ideally, air ducts in the sound room should not be shared with other rooms. Further, if during construction they are made of duct board, a compressed fiberglass, much of the sound that gets into the ducts will be dissipated before it travels to other rooms.
2. Insulation
   Fiberglass insulation and the like has only a minor effect on sound, often deemed not worth the effort or expense in view of all the more important techniques. Further, if stuffed tightly, the insulation can actually transmit sound from one side of the wall to the other.

Resources:
www.soundproofing101.com
www.soundonsound.com/sos/jul00/articles/aqacoustic.html
www.acoustiblok.com
www.soundproofing.org