- How are you feeling today?
The second sentence, does it have a name, is it a one-word response?
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It's called a sentence word or a one-word sentence:
A sentence word (also called a one-word sentence) is a single word that forms a full sentence.
As already answered, the correct English term for a complete sentence composed of just one word is a sentence word.
But there are other more creative ways you could describe such a statement.
"Monosyllabic" describes both a single syllable statement and can be used to describe a person who speaks in such a way.
In novels, verbs such as "grunted" may be used to describe the speech someone who spoke in single syllable utterances; for example:
"How are you feeling", asked Sue.
"Bad", grunted Bob.
A sentence can be defined as:
a word, clause, or phrase or a group of clauses or phrases forming a syntactic unit which expresses an assertion, a question, a command, a wish, an exclamation, or the performance of an action, that in writing usually begins with a capital letter and concludes with appropriate end punctuation, and that in speaking is distinguished by characteristic patterns of stress, pitch, and pauses
However, it's more common for it to be thought of in the following way:
In grammar, a sentence is the basic grammatical unit. It contains a group of words and expresses a complete thought.
A sentence consists of a subject and a predicate. For example in the sentence "Bill writes good poems" Bill is the subject of the sentence and writes good poems is the predicate.
The one-word sentence No seemingly contains none of those things.
However, those things are both implied by and inferred from the context of the sentence that came before:
"How are you feeling today?"
"(I'm feeling) Bad (today)."
In this case, No is known as an elliptical sentence, where one or more words have been omitted yet are assumed to exist in context. (Omitting all but a single word is an extreme form of elliptical construction.)