Consider the question:
What are the correct terms for a concept and the real thing it represents?
This answer won't attempt to provide "correct terms" since philosophers may differ about what is correct. Rather I will focus on how Gottlob Frege defines these concepts in On Sense and Reference.
Frege is concerned about equality and knowledge:
Equality gives rise to challenging questions which are not altogether
easy to answer. Is it a relation? A relation between objects, or
between names or signs of objects?
He starts off with "signs" or "names" and expresses equality by a=a or a=b.
Now a=a can be established a priori, but a=b may not and because of that may offer "very valuable extensions of our knowledge".
What the sign refers to Frege calls its "reference". On a superficial level the answer to the OP's question may be simply that the "concept" is called the "sign" and the "real thing it represents" is called the "reference".
However, there may be more that needs clarification especially if one is also interested in what equality means and how we may extend our knowledge. Different signs may have the same reference. He provides an example of this by using the midpoints of triangles:
Let a, b, c be the lines connecting the vertices of a triangle with
the midpoints of the opposite sides. The point of intersection of a
and b is then the same as the point of intersection of b and c. So we
have different designations for the same point, and these names
('point of intersection of a and b', 'point of intersection of b and
c') likewise indicate the mode of presentation; and hence the
statement contains actual knowledge.
In that example there were two ways to designate the same referent that at least in their mode of presentation are different. He calls these designations the "sense of the sign":
It is natural, now, to think of there being connected with a sign
(name, combination of words, letter), besides that to which the sign
refers, which may be called the reference of the sign, also what I
should like to call the sense of the sign, wherein the mode of
presentation is contained. In our example, accordingly, the reference
of the expressions 'point of intersection of a and b' and 'point of
intersection of b and c' would be the same, but not their senses. The
reference of 'evening star' would be the same as that of 'morning
star', but not the sense.
So we need three terms to connect a concept and the real thing it represents.
- The sign we use in our language.
- The sense of the sign.
- The reference which the sense points to which may have many senses and signs which can be connected by an equality relationship offering us extensions of our knowledge.
Frege, G. "On Sense and Reference", translated by Max Black, Wikisource https://en.wikisource.org/wiki/On%2520Sense%2520and%2520Reference?oldid=4276834
Wikipedia, "Sense and reference" https://en.wikipedia.org/wiki/Sense_and_reference