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1.
A small dog possessing one or more of the following characteristics: bulgy eyes, squashed face, eyes that look in two or more directions at the same time, wandering lung, or comically vacuous expression. In short, any dog that arouses feelings of pity, revulsion, and the sneaking suspicion that it would have been kinder to have strangled it at birth.

E.g. pugs, Boston terriers, French bulldogs.
For the origin, do a google search for "hurr durr derp derp derp"
Is that a derp-derp or is she just squeezing her dog too hard?
by Boston Terrierist May 10, 2009
 
2.
Nickname of the Sea Otter Water-type starter for Pokemon Black/White on the GameFAQs Pokemon Black/White boards, so named due to its goofy, uncompelling design, especially compared to his Grass and Fire counterparts. Described as "the bastard child of a Jynx and a Bidoof" by many GameFAQs forumers.
Derpderp is only sad because he wishes he were a Smugleaf
by agnryman May 27, 2010