Question:

Can you tell me what the vow was that caused Paul to cut his hair? 

Answer:

"So Paul still remained a good while. Then he took leave of the brethren and sailed for Syria, and Priscilla and Aquila were with him. He had his hair cut off at Cenchrea, for he had taken a vow" (Acts 18:18).

The shaving of the head in connection with a vow is following the custom of the Nazirite vow (Numbers 6:2-21). So we know the type of vow that Paul took, but the Scriptures do not record for us what the particular vow Paul made was about.