The person that Romeo asks to marry him and Juliet is this priest named Friar Lawrence.  Friar Lawrence seems to be some sort of a spiritual adviser to Romeo.  We see that he has given Romeo advice on various things.  For example, we know that he has been talking to Romeo about his relationship with Rosaline and about love in general.

Later on, Juliet will trust Friar Lawrence enough to take the potion that will make her seem to be dead.  So clearly, the two of them trust him a lot.  Friar Lawrence agrees to marry the two because he hopes their marriage will make peace between the Capulets and Montagues.