Soundproofing

Flooring

One of the best ways to keep sounds from going between units is to use flooring materials that reduce the transmission of sounds. Floors are rated with two numbers. 

The combined STC (Sound Transmission Class) rating for the new floor system shall be at least 58 IIC. The combined IIC (Impact Isolation Class) rating for the new floor system shall be at least 55 STC.

Seal Your Unit 

Caulk or seal any cracks or holes in your walls.  Use insulators to seal around outlets & switches.

Why Insulate Electrical Outlets in your HomeOutlet Insulation Stops Cold Air Coming Through Electrical Outlets (DIY) | Family Handyman

Add Insulation

Buy Sound Batt Insulation | Insulation ...

Add sound rated insulation in your ceilings and walls if you do work on either.