A degree adverb is an adverb which is used to modify the degree or strength of the adjective that comes after the adverb. There are two types of degree adverbs. The first type is intensifiers and the second type is downtoners (DeCapua 107).
Intensifiers are adverbs which strengthen the degree of adjectives they modify. The modified adjectives must always be gradable, which means that they must have the comparative and superlative forms. The most common examples of intensifiers are: extremely, amazingly, terribly, marvelously, completely, fully, etc.
For instance, let us take a look at the following sentence - "The boys were extremely quiet." The phrase "extremely quiet" consists of an adverb which modifies the adjective that comes after it. The modifying adverb is an intensifier because the degree of the adjective "quiet" is strengthened. The boys were not just quiet, they were "extremely" quiet, which means that they were very quiet.
Downtoners are adverbs which mitigate the intensity of the adjectives they modify. Just like intensifiers, they modify the gradable adjectives. Common examples of downtoners are: fairly, somewhat, barely, hardly, scarcely, etc.
For instance, we can examine the following sentence - "The movie was somewhat boring." The word "somewhat" is a downtoner whose purpose is to lessen the intensity of the adjective "boring." So, this sentence means that the movie was not completely boring, only to some extent.
DeCapua, Andrea. "Different Subclasses of Adverbs." Grammar for Teachers: A Guide to American English for Native and Non-native Speakers. New York: Springer, 2008. 107-08. Web. 24 Nov. 2015.