A sound wave can be controlled in one of the three ways where it can be reflected, diffused or absorbed. Depending on the nature and composition of material it comes in contact with, each can be used to some extent in soundproofing.
Reflection: Sound is bounced off a surface. It normally occurs on flat, rigid surfaces with a lot of mass like concrete or brick walls. As the sound wave can’t penetrate very far into the surface, the wave is turned back on itself like a ricochet. The sound bouncing back off the surface creates an echo.
Diffusion: When a sound wave hits an irregular surface like foam or carpet, the vibration breaks up and travels along many much smaller paths. This divides the energy of the wave, sending it in many different directions which depletes its energy faster.
Absorption: When a sound wave hit a particular surface, the kinetic energy driving it is converted into a small amount of heat energy which dissipates, leeching power from the sound wave and causing it to decaying faster. This is the kind of sound insulation provided by things like foam and rubber. How well a material absorbs sound depends on a number of different factors, including material density and how porous it is.