peekThere doesn't seem to be much difference between peek and peep, but there is a difference between peer and the other two.
peek - Dictionary definition and pronunciation - Yahoo! Education
To peek is to glance quickly or look or peer furtively as from a place of concealment.
peep (1) definition - Dictionary - MSN Encarta
To peep is to look quickly or secretly (for example) through a small opening or hiding place.
peer - Dictionary definition and pronunciation - Yahoo! Education
To peer is not only to look but to look searchingly, intently or with difficulty.