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To make this easier, let's put "swiftly into a sentence:
The trees moved swiftly with the grace of the wind.
First we need to know what verbs, nouns, adjectives, and adverbs are:
A noun is a person, place, or thing. "Swiftly" is not any of these. The nouns in this sentence are "trees", "grace", and "wind."
A verb is the action of the noun/pronoun, or what the noun/pronoun is doing. "Moved" is the verb in this sentence, because the trees are completing the action of moving. So "swiftly" is not a verb.
An adjective is something that describes the noun. There are no adjectives in this sentence because the "trees", "grace", and "wind" are not being described.
An adverb is a word that describes the verb, usually ending in -ly. Aha! The -ly rule should automatically give you your answer. "Swiftly" is an adverb. Also, the word "swiftly" describes how the trees are moving, and it even comes right after the verb. Therefore, "swiftly" is an adverb.
