What are the different kinds of nevus, and are certain kinds at high risk for turning cancerous?

Dozens. There are dozens of nevi -- although compound nevus and intradermal nevus are the most common acquired nevi. A dysplastic nevus (clark nevus) has the potential to be melanoma in a limited number of cases. In general, though, nevi are not pre-cancerous lesions.
Benign,dysplastic,ca. A dermatologist (skin doctor) is trained to differentiate among types of moles, also called nevi. That expertise has been shown to be live saving. A short overview is at http://en.Wikipedia.Org/wiki/dysplastic_nevus . Some people tend to have multiple dysplastic (abnormal appearing) nevi, and some have a genetic syndrome. The risk of being or becoming cancer varies type of nevus.