If pathogens pass the non-specific first line of defence they will cause an infection. However, the body has a second line of defence to stop or minimise this infection. This is called the immune system. As a part of this there are two types of white blood cell called phagocytes and lymphocytes.
Phagocytes surround any pathogens in the blood and engulf them. They are attracted to pathogens and bind to them.
The phagocytes membrane surrounds the pathogen and enzymes found inside the cell break down the pathogen in order to destroy it. As phagocytes do this to all pathogens that they encounter, they are called 'non-specific'.
Lymphocytes are another type of white blood cell. They recognise proteins on the surface of pathogens called antigens. Lymphocytes detect that these are foreign not naturally occurring within your body and produce antibodies. This can take a few days, during which time you may feel ill. The antibodies cause pathogens to stick together and make it easier for phagocytes to engulf them.
Some pathogens produce toxins which make you feel ill. Lymphocytes can also produce antitoxins to neutralise these toxins. Both the antibodies and antitoxins are highly specific to the antigen on the pathogen, thus the lymphocytes that produce them are called 'specific'.