What function do they serve?  

The tonsils and the adenoids are mostly composed of lymphoid tissue, which is found throughout the gastrointestinal tract and on the base of the tongue.  Lymphoid tissue is composed of lymphocytes...which are mostly involved in antibody production.

Since we generally consider antibody production to be a good thing, many studies have been performed to try to clarify the importance of the tonsils.   There seems to be no adverse effect on the immune status or health of patients who have had them removed.  Any noticeable effect has generally been positive. It appears that the tonsils and adenoids were not "designed" to effectively handle the multitude of viral infections that occur in children in an urban population.  Rather, the immune system, including the tonsils and adenoids, developed during a era where the child was rarely exposed to a large number of other people and the germs they carried.  It may also be that these organs are relatively more important in dealing with certain types of infections, such as worms or other parasites, that are relatively uncommon in today's society.  It is clear that in many cases, the tonsils and/or the adenoids become "dysfunctional" and are more of a liability than an asset.