Sound waves will spread to any place there's an opening into your home or spots where sound-deadening materials aren't sufficient. The major culprits for this problem are doors, windows and ductwork systems. Fortunately, increasing the sound reduction capabilities of these surfaces is relatively cheap and simple.
Many apartment dwellers complain about noise from the hallways passing into their homes. To solve this problem, residents can invest in STC-rated doors. The average hollow-core apartment door has an STC rating of between 20 and 25 [source: CR Mayer]. This means that even the average conversation taking place in the halls can be heard inside of your apartment with relative clarity. STC doors are available at ratings as high as 55 [source: Krieger]. The drawback to STC-rated doors is that they're fairly heavy and expensive, and may require special hardware beyond that currently on your door. Many apartments may not even allow tenants to replace their doors.
For a simpler and more affordable way to keep noise from coming through your doorways, consider the use of sound seals. These are neoprene or vinyl strips that are installed around the door to seal any open spaces. Cheap stick-on versions are effective, but for a longer-lasting solution, look for seals that can be screwed in place. They are a bit more expensive, but they won't keep falling like stick-on types can.
To further seal your door, add a metal threshold and surface-mounted door bottom seal to the door. Both of these items can be installed within a few minutes, but will greatly reduce the amount of noise that enters from under the door.