Q. Is it true that a kohen may not marry a convert?
A. The answer is yes, but this is no reflection on the convert. The kohanim have serious ritual responsibilities and are subject to restrictions as to what they can do (e.g. they must keep away from contact with the dead unless it is one of their close relatives) and whom they may marry (not a divorcee, not a convert, etc.).
This is no reflection on the convert; indeed Ruth, the role model of Biblical converts, was a person of great strength of character and exemplary ethics.
There are some recorded cases where a kohen was in fact given rabbinic permission to marry a convert, but these marriages took place in highly unusual circumstances, were criticised by the majority of rabbinic authorities, and are not recognised as halachic precedents.