As a professional number theorist, I would not say that this is accurate.
A representation of a group, formally, is a homomorphism of that group to the group of automorphisms of some vector space. More informally, it is a way to make the group "act on something".
Here is an example. Consider the group generated by the symbols a, b, and their inverses a^{-1} and b^{-1}. As usual write a^2 = aa, a^0 = 1, etc. And impose the relations
a^4 = b^2 = 1, a^3 b = ba.
At first, this is very hard to understand. Which group elements are equivalent to which others? Is the group finite or infinite? If finite, how many elements does it have?
To construct a "representation", make a square out of cardboard, mark the edges and sides, and interpret the symbols in the following way: a means rotate clockwise 90 degrees, and b means flip across its vertical axis. a^{-1} and b^{-1} mean do the same thing in reverse.
Now you can see, for example, that this group has exactly eight elements -- corresponding to the positions of the cardboard you can reach.
For any group, there are always representations which you can easily construct, such as this one:
https://en.wikipedia.org/wiki/Regular_representation
But there are representations which are "hidden" in some sense, where it wasn't clear initially that they should exist at all. That's where the magic happens.
The Langlands program is extremely technical; I specialize in another area of number theory, and I only vaguely understand it myself. But very very roughly speaking, the Langlands program describes multiple ways of constructing certain kinds of group representations, and says that you end up constructing the same representations.
The modularity theorem, which was the linchpin in the proof of Fermat's Last Theorem, is an example of a theorem along these lines.
https://en.wikipedia.org/wiki/Modularity_theorem
And the Langlands program is very far from complete.