A diminutive or insignificant person, especially a sprightly or impertinent youngster.
'Whipper snapper' is now a rather archaic term and, although you might hear it in black and white British films, those who are young and streetwise enough to actually be whipper snappers aren't likely to use it.
'Whipper snappers' were known by various names, all of them derived from the habit of young layabouts of hanging around snapping whips to pass the time. Originally these ne'er-do-wells were known simply, and without any great linguistic imagination, as 'whip snappers'. This term merged with an existing 17th century term for street rogues - 'snipper snappers', to become 'whipper snapper'. Christopher Marlowe mentions 'snipper snapper' in the 1604 edition of The Tragicall History of Doctor Faustus, when referring to a 'hey-pass', which is what street jugglers were known as in Marlowe's day.
But I'll seeke out my Doctor... O yonder is his snipper snapper... You, hey-pass, where's your master?
The meaning of 'whipper snapper' has altered over the years, originally referring to a young man with no apparent get up and go, to be applied to a youngster with an excess of both ambition and impudence.
See also the meaning and origin of 'whipping boy'.