For a person to be "an ass" is not well-defined. Generally, it suggests that the person consistently behaves offensively, although it can also be applied to a specific incident.
The word "ass" actually has two separate meanings/etymologies http://www.etymonline.com/index.php?term=ass On the one hand, the oldest derogatory meaning likens a person to an obstinate, stupid animal, the ass (or donkey), as in the phrase "he made an ass of himself", and the connection goes back at least to Shakespear's "A Midsummer Night's Dream". The fairy Puck considers the inept actor Bottom to be an ass, and therefore changes his head to that of an ass, literally "making an ass of him".
The second, more recent, line of development, starting with the British "arse", changed it to "ass", which is to say the part we all sit on. Since both meanings are unsavory (in certain social circles), the two meanings have to some degree become mingled when referring to objectionable behavior, and in contemporary American usage, the derogatory "ass" is usually shorthand for the more offensive "asshole".
There was a young girl from Madras
Who had a magnificent ass.
It was not, as you think,
Round, firm and pink;
It was grey, had long ears and ate grass.