‘Emphatic sentence’ is not a formal grammatical term, just a loose description. It appears that in this particular case your teachers have used it as user-friendly term for what grammarians call an It- cleft. That kind of sentence is employed to restructure a simple sentence like ...
You are after my black cat.
... in a manner which puts the most important point in the ’new information’ position immediately after the verb. In this case, the sentence is “cleft”—cut into two pieces:
You are after | my black cat.
The original front half, You are after, is moved to the end and subordinated as a relative clause, which you are after:
my black cat | which you are after
A new subject is introduced: the ‘dummy’ pronoun, it, which has no referent.
It .. my black cat which you are after
This is joined with BE to the original back half, which is now in the desired emphatic position.
It is my black cat which you are after.
Your second sentence is not a cleft. That is the ordinary demonstrative pronoun, acting as subject. Its referent is black cat.
Formally the sentence may be parsed thus:
[subject NPThat] [verbis] [complement NPmy black cat [relative clausethat you are after]].
You are correct, however, in suspecting that this represents a re-arrangement of a simpler sentence. What has been ‘cleft’ here is the NP That black cat that you are after; the underlying sentence is
[subjectThat black cat [relative clausethat you are after]] [verbis] [complement pronounmine].
In order to put the emphasis on the ‘my-ness’ of the cat,
We move the very heavy subject NP to the end, leaving behind the fairly trivial determiner which heads it, that:
That _ is mine | black cat that you are after.
That, now at the head of the sentence, is recast as a pronoun and acts as subject:
That is mine ...
We rejoin the two parts by recasting the original complement pronoun mine as a possessive adjective:
That is my ...
This puts the term the speaker wants to emphasize at the head of the complement—the ‘new information’ position—and permits it be spoken with the primary emphasis:
That’s my black cat ...