example sentence: The crowd reacted with vigor after the team scored a goal.
This means they reacted with strong feelings, and excitement, in this case.
Vigor has a few different definitions. It is a noun. It is used to describe other nouns or actions. It generally means strength; physical strength, intellectual strength, strong emotions or reactions, etc.
It is commonly used to describe another noun or verb by adding "with vigor"
Vigorous is the adjective (noun descriptor) form, and vigorously is the adverb (verb descriptor) form.
I hope this helps!