A Nazarite was someone who took a vow. While under that vow, they were to refrain from alcohol of any kind, they had to stay away from dead bodies, and they couldn't cut their hair. These stipulations were in effect until their vow was fulfilled. Some Jews took a Nazarite vow to see to the death of Paul. They vowed not to eat until Paul died. That turned out to be a long fast. Samson, in Judges, was a Nazarite. This was one of the reasons he lost his strength when Delilah cut his hair, for he didn't fulfill his promise to God. It is thought that John the Baptist may have been a Nazarite, for you may remember the Angel Gabriel telling Zachariah that no alcohol would pass his lips and from his description, not too many scissors passed over his scalp either! And in Acts, roughly chapter 21, Paul paid for the fulfillment rituals of 4 Nazarites to appease the Jews who thought Paul had lost his Jewish identity.
A Nazarene is someone from Nazareth.