Soundproofing is made up of two main components, air and barriers. Sound travels through the air until it comes up against a barrier that lowers its intensity, the more times the sound comes upon a barrier the more power is lost. Low density materials work very well to interfere with sound waves and reduce the energy within air pockets of the material. Styrofoam and home insulation products are a good example of this, very light weight with a lot of room for air and also can be made into pieces large enough to cover walls and ceilings relatively cheaply. Multiple layers of insulation can be used for better results, but be sure to make space between the sheets for added efficiency. However, you can have many layers of insulation but if they are connected with materials that are too rigid then that may reduce the usefulness of the structure.
Sound also travels through solid material. This sort of energy is generally a lower frequency like a kick drum or a diesel engine. To prevent this type of audible transference it’s best to use materials lower in stiffness and rigidity. Metal can easily transfer sound energy through a long distance, for example a bystander near a railroad may hear evidence of an oncoming train miles away through the vibration of the metal tracks. When constructing a soundproof space it’s best to use material that will dampen the force of the sound through that material. Wood is widely available and has decent sound dampening qualities. Certain polymers made for soundproofing should be sought after because manufacturers are able to create very stable construction materials out of plastic that are also lightweight and can have a controlled density to aid in volume reduction.
It’s a good idea to make your soundproofing space as irregularly shaped as you can. Sound likes to travel in one direction so every time it’s sent on a different path it loses energy. If, for example, you’re setup to record in a normal 10’ x 10’ room that had all walls 90° from each other, the sound will bounce back toward the microphone so much that it would compromise quality. Purchasing some large pieces of foam and randomly placing them around that same room would dramatically reduce the amount of indirect sound making its way back to the mic. The more work you put a sound wave through, the more energy you take from it thereby providing the best soundproofing possible.