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1.
A adjective to describe someone who is cute (not in the sexual way) or duckyshun-like. Usually describes overweight/slightly chubby women, Disney characters or animals with fur and a 'squirilly' (as in squirrel-like) nature.
'Mad Madam Mim is so cudgy!'
'That duck is so cudgy'
by Sharon Duckshun August 12, 2006
 
2.
Cute but edgy.
Boy1 : "Have you seen the math teacher's new hair cut?"
Boy 2 : "Yeah! It's cudgy!"
by Toodark October 07, 2010