We immediately discover that Brown thinks of the errand as having an "evil purpose." As he meets with the mysterious shadowy figure who guides him, he states his belief that the evil in which he is participating is one that neither his family, nor his community would countenance, a statement that is disproven when he finds the entire town, including his wife, participating in dark rituals in the woods. 

His purpose is not directly stated, but, as this is a strongly religious story, we should view it theologically. It seems that curiosity is a driving force as is the influence of the Devil. Given the Puritan background of the story, we can argue that all men are naturally susceptible to temptation (and are fallen due to Original Sin). Salvation is only possible through faith, and yet, in rejecting Faith's request that he not go on the journey, Brown gives up his one defense against the inherent fallen nature of humanity and thus falls prey to the Devil.  

Young Goodman Brown tells his wife, Faith, that he must go to the forest tonight, that it must be accomplished between "'now and sunrise.'"  After he's left home, he looks back and sees her peering down the road after him, looking sadly, and he feels guilty for leaving her to go on "'such an errand.'"  The narrator describes this errand as his "present evil purpose," and such a description makes it sound as though Brown has had other evil purposes in the past.  Further, he clearly seems to know the old man he meets in the forest, and this would indicate that he has met him before. 

It seems as though Brown's intention is to commit some sin, some transgression for the last time, as his own admission is that "'after this one night, [he'll] cling to [Faith's] skirts and follow her to Heaven.'"  He seems to be aware that he will commit some sin in the forest because he intends to be good, so to speak, starting tomorrow.  Whatever this sin is, it is never explicitly stated, but we get quite a few hints that this was his plan when he left home.