Is epsilon error a standard known error or custom created by this paper?
What is that equation and is it standardly used?

I havn't seen exactly that before, so I guess it's not &quot;standard&quot;, but it's not so unusual.  The equation is &quot;just&quot;  1 - (normal distribution).  If you think of the normal distribution as a measure of &quot;how similar a sample is to the mean&quot;, then that equation turns the similarity (big is good) into a distance (small is good).
It looks related to &quot;robust loss functions&quot;;  See below.

What is x in this equation? I thought it was some type of feature
scaling of the image but that makes no sense because the results are
all different in the table and completely out of context anyway. Is x
supposed to be the predicted age?

Your second thought is right.  $x$ is predicted age.  They use that set up since they don't have a single correct answer for their data, but rather, many guesses/votes as what the correct age is.
From the paper:

The LAP challenge evaluation employs fitting a normal distribution with the mean  μ and standard deviation σ of the votes for each image.

Say you have two images. For one, every guess you have for the age is the same.  For another, you get a wide range of guesses for the age.  The second image must be &quot;harder&quot; somehow.  This equation gives you an error measure that tells you how the algorithm performed relative to the many people that guessed the true age.
Robust losses
Check out this work
The &quot;epsilon-error&quot; equation from the paper you linked to will look sort-of like the bottom (orange) trace in Figure 1: