In short, Nazirites were people who vowed to set themselves apart to the Lord for a time.
The original definition of Nazirites is the description Lord gives to Moses in Numbers 6. I think it makes sense to stick to the original, so I'll provide the relevant passage here, highlighting the points that set Nazirites apart.
Numbers 6:1-12 (NLT) (emphases mine)
1 Then the Lord said to Moses, “Give the following instructions to the people of Israel.
2 “If any of the people, either men or women, take the special vow of a Nazirite, setting themselves apart to the Lord in a special way, 3 they must give up wine and other alcoholic drinks. They must not use vinegar made from wine or from other alcoholic drinks, they must not drink fresh grape juice, and they must not eat grapes or raisins. 4 As long as they are bound by their Nazirite vow, they are not allowed to eat or drink anything that comes from a grapevine—not even the grape seeds or skins.
5 “They must never cut their hair throughout the time of their vow, for they are holy and set apart to the Lord. Until the time of their vow has been fulfilled, they must let their hair grow long. 6 And they must not go near a dead body during the entire period of their vow to the Lord. 7 Even if the dead person is their own father, mother, brother, or sister, they must not defile themselves, for the hair on their head is the symbol of their separation to God. 8 This requirement applies as long as they are set apart to the Lord.
9 “If someone falls dead beside them, the hair they have dedicated will be defiled. They must wait for seven days and then shave their heads. Then they will be cleansed from their defilement. 10 On the eighth day they must bring two turtledoves or two young pigeons to the priest at the entrance of the Tabernacle. 11 The priest will offer one of the birds for a sin offering and the other for a burnt offering. In this way, he will purify them from the guilt they incurred through contact with the dead body. Then they must reaffirm their commitment and let their hair begin to grow again. 12 The days of their vow that were completed before their defilement no longer count. They must rededicate themselves to the Lord as a Nazirite for the full term of their vow, and each must bring a one-year-old male lamb for a guilt offering.
The passage goes on to lay ritual law for when the person's time as Nazirite ends.