Sophie is correct. Perverse is an adjective and pervert is either a verb or a noun. Note that the pronunciations for the verb and noun are slightly different (here at least). For the noun, the stress is on the first syllable, and for the verb, it's on the second.
The verb means to twist, or misinterpret, and usually is used in the idiom 'pervert the course of justice', which refers to giving false evidence, or influencing witnesses.
The noun usually refers to a person whose sexual tastes or behaviours do not align with those of the society they live in. 'Watch out for that old pervert. He likes to grope little boys'.
Perverse is used to describe behaviours and tastes, but is also sometimes used in non-sexual contexts, e.g. 'His musical taste is perverse! I heard him singing Abba songs in the shower'.