Pro-forms
A pro-form is a word that can be used as a replacement to avoid redundancy without removing meaning from a sentence.
Its best known example is the pronoun.
With some mental flexibility, you can also see pro-verbs and even pro-sentences - consider "Yes.".
For adjectives and adverbs it is harder to demonstrate.
Pronoun
A specific kind of pro-form, a pronoun (adj: pronominal) replaces a noun or noun phrase with a simpler word, often to avoid repetition and shorten a sentence for ease of speaking.
I, you, them, he, she, they.
For example, "Paul ate the apples, then he threw them away."
Writing that as "Paul ate the apples, then Paul threw the apples away" would be grammatically correct, and semantically identical in that it explains the same thing, yet it would feels oddly verbose because of its closeby redundancy, and we generally avoid that redundancy by using pronouns.
...to the point that not using pronouns quickly starts feeling feels like something like rhetoric anaphora (emphasis by repetition).
Since pronouns can refer to things in the current, earlier and even later sentences, a pronoun should have only one sensible referent, and the writing should be aware of all the possible referents.
References can be made clearer by using some type of agreement. The previous example uses grammatical gender and number. In reality this is often a process disambiguation that relies on a reader's 'does this make any sense?' sense.
Pro-verb
In "Can I kick it? Yes, you can.", you can say that 'can' is a replacement of the previous sentence. Or that the omission of a subject implies that reference.
The second sentence is understood as "Yes, you can kick it."
This runs right into how anaphora work, in particular around omissions.
It itself is also a pro-form, specifically a pronoun, a reference to whatever it is we are supposed to be kicking.
Pro-sentence
A pro-sentence refers to one or a few words referring to an entire sentence or a large part of it.
In english, an extreme case is the answer "Yes." It's not a valid independent sentence, but an an answer to the first sentence before, meaning something like "Yes, you can kick it."
This could be seen as a forceful stretch preferred by those for whom analysis depends on sentences having all basic parts. In discourse analysis - which is probably the most useful for uttered dialogue anyhow - this is not necessarily necessary.