What is the 'sense' of a word?

That would be the word's connotation.

Explanation:

Connotation - the way a word sounds, the 'sense' that it gives

Ex: the difference between saying 'weird' and 'quirky'

Denotation - the word's definition

So while 'weird' and 'quirky' have the same definition (or denotation), 'quirky' sounds a bit better, so they have different connotations. The 'senses' are a bit different.