In this context, the PES is not proper of a reaction but the nuclei involved.
You should take in mind that, in absence of external fields (and taking apart spin), it is just a functional of the nuclei positions, and as there are many ways to describe nuclei positions there are many ways of visualize the PES.
For a "large" number of nuclei, the dimension of the PES will be very large. Ideally, 3$N$-6 with the considerations above.
How it looks like? It is hard to imagine and represent in an 2D screen. By the way, I never found it useful, no more than to imagine that the reaction is a ball rolling down a hill.
To turn it into something that can be useful, the PES is projected into a subspace spanned by the vectors linked to the coordinates of interest for the case (for example, two bond distances).
Does it answer your question?
Edit:
- Why make a distinction between a 'reaction' and the movement of the system's nuclei
It could be done, but with "the PES is not proper of a reaction but the nuclei involved" I meant that the whole PES is independent of the trajectory followed by the nuclei during the reaction.
- Can you show me an example of a projection of a PES? (Or do you just mean the PEP, as it is shown in most computational journals?)
I mean the projections of the PES usually plotted. With PES is projected into a subspace spanned by the vectors linked to the coordinates of interest for the case I meant the following. The whole functional cannot be plotted in high dimensional cases. So, the trick is chose a set of variables that allow the description of the nuclei positions BUT also allow just use a few to describe the important part (which depends on your needs).
A simple example would be the PES corresponding to the model of a benzene molecule (built for teaching purposes) in a laboratory when you are interested in teaching parabolic movement. Of course you can describe it with Cartesian coordinates of each nuclei in R^3 (12*3=36 coordinates). But they are not very useful, you need to make appear the $x^2$ somehow for your young students. So you make a change of basis and get a new set of coordinates. Three translational of the geometrical centre and internal coordinates.
Then, you can make a matrix representation of this vector:
[int1, int2, ..., int33, z, x, y]$^T$.
It is horrible to write here (you'll see why soon) so fortunately found a little ball. And you reduce it to
[z, x, y]$^T$
But your student have to plot the famous parabolic case in a paper, and you only need $x$ and $y$, so you can just discard $z$ or make a projection by applying the matrix representation of an operator $P= P^2$.
$$ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \left[ \begin{array}{c} z \\ x \\ y \end{array} \right] = \left[ \begin{array}{c} 0 \\ x \\ y \end{array} \right]$$
now is simpler. The change is subtle (maybe pedantic), one case is an approximated plot of something exact, the second one is an exact plot of something approximated. By the way, the plot is the same...
My advise is: Just look at the PES as a simple function $R^N \rightarrow R$ with $N$ at least as 3$n$-6. There is nothing special about it that makes it different from what you studied in your algebra/calculus classes. The only one particularity is what the variables represents (nuclear positions), it was well described by the @R.M. answer.
I really never find it too useful more than a fast inspection of simple energy curve. I found the mathematical machinery of the optimisation algorithms much more useful. But of course, I looks very nice in papers, I would recommend a good colour scheme.