Connote usually means the implication behind a word. For instance, if you compare the two words 'argue' and 'persuade,' argue has more negative connotation because it brings to mind images of someone red in the face yelling. The word persuade has a calmer connotation and also implies a one-sidedness that the word argue does not.
Included below is the dictionary.com definition of the word 'connote'.
http://dictionary.reference.com/browse/connote