What part of the immune system is responsible for destroying or neutralizing pathogens that have entered the body?

Macrophages kill the cellular debris of the pathogens that are killed by the immune system. They are a type of white blood cells, and kill anything which does not have the proteins that are specific to the healthy cells of the body. Cancer cells, foreign substances, and microbes are engulfed by them.

The part of the immune system that destroys pathogens are the antibodies.
Antibodies are large protein molecules, each having a specific shape and binding site, similar to enzymes, which bind to the surface antigens of the pathogen and neutralize them. There a number of mechanisms through which antibodies achieve this, such as rupturing the cell membrane of the pathogen.
Antibodies also lead to immunity, if the antibodies to a certain pathogen are present, then the organism is immune to that pathogen.