According to the OED, an adverb is the:
Name of one of the Parts of Speech: a word used to express the attribute of an attribute; which expresses any relation of place, time, circumstance, causality, manner, or degree, or which modifies or limits an attribute, or predicate, or their modification; a word that modifies or qualifies an adjective, verb, or other adverb. Also used attrib.
Because the distinction between prepositions and adverbs in English can be somewhat blurry, on certain occasions adverbs can modify words that are under most analyses acting as prepositions.
Adverbs are not limited to words ending in -ly. Furthermore, they inflect into the comparative and superlative under the same rules used for adjectives:
- to speak honestly / more honestly / most honestly
- to run fast / faster / fastest
However, just as nouns used attributively are not subject to an adjective’s comparative and superlative degrees (like floor boards), nouns used adverbially like tomorrow or yesterday are also not subject to inflections of degree.
See Wikipedia’s page on adverbs for more information about this class of word.