A person taking a Nazarite vow was to abstain for a specific period from partaking of grapes or any of its products whether intoxicating or not, cutting his hair, and touching a corpse . Such a person is called a Nazirite (Heb. nazir, נָזִיר). At the end of the period of abstention the Nazarite shall shave his head and put the hair in the fire which is under the sacrifice of the peace offerings. If a nazirite fails in fulfilling the three obligations there may be consequences, such as that all or part of the person's time as a nazirite may need to be repeated. The subject is dealt with in the Priestly Code, in Numbers 6:1–21.
Ellicott's Commentary for English Readers says there can be no doubt that the “vow” was that of the temporary Nazarite, as described in Numbers 6:1-21. Ellicott's says the grammatical structure of the Greek sentence makes it possible to refer the words to Aquila as well as St. Paul, but there is hardly the shadow of a doubt that the latter is meant. So, Acts 18:18 is telling us that Paul had taken the Nazarite vow, which would be fulfilled by burning his hair on the temple altar.
Avram Yehoshua (The Lifting of the Veil: Acts 15:20-21, page 200 note 563 ) says that scholars are perplexed that Paul would do such a thing. Yehoshua says (page 201) anyone taking the vow held God's law in the highest esteem. Although a proud Jew, Paul, in his own epistles, frequently referred to Christians as not under the law of the Jews, thus making this vow seem incongruous for him. This could not be more obvious than in the following passage:
Romans 6:14-15: For sin shall not have dominion over you: for ye are not under the law, but under grace. What then? shall we sin, because we are not under the law, but under grace? God forbid.
Before dealing with why Paul took the Nazarite vow, we should first attempt to establish whether he did actually take the Nazarite vow.
Mark's Gospel several times refers to Jesus as a Nazarene, and Matthew 2:23 acknowledges this appellation, but Jesus was not a Nazirite and is never described as one. Acts 24:5 has Paul described as being a Nazarene, clearly an accusation that could not have been understood as 'Nazarite', but in answering that accusation, Paul says that Jews found him purified in the temple (Acts 24:18), an allusion to the process a Nazarite underwent. Luke appears to be deliberately conflating the two terms.
Cenchrea is in Corinth, where Paul had to this point continuously spent more than a year and a half among the gentiles (Acts 18:8,11,18). In his epistles, Paul tells the gentiles that they are not under the Jewish law, yet three times in Acts, he contradicts this by taking the Nazarite vow in the presence of gentiles, who are precluded from the same vow, yet in 1 Corinthians 11:1 asks them to be his imitators (μιμηταί). It is inexplicable that Paul would slight the Corinthians in this way, and inexplicable that many were converted.
Acts could be seen to portray Paul as a Nazarite in three separate passages: Acts 18:18;21:24-26;24:18. The author sometimes uses triple repetition to persuade his readers regarding the veracity of an event. Another example is the disputed conversion on the road to Damascus (i), described in Acts 9:1-19;22:4-16;26:12-18. Of course, triple repetition does not disprove any of these events, but it is specific to Luke's style. If scholars are perplexed that Paul would do such a thing, it is possible that Luke is portraying Paul as taking the Nazarite vow for the author's own theological reasons.
(i) Some scholars, including Uta Ranke-Heinemann, have noted that the conversion accounts appears to have been based on the Bacchae, a play by Euripedes. Also see Paul's Conversion/Call: A Comparative Analysis Of The Three Reports In Acts for a discussion of the conversion accounts as " theologically and stylistically motivated" elements of Luke's literary technique.