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Imagine there's a random variable such as $ε$. Then we say that $ε$ is i.i.d and follows a normal distribution with mean $0$ and variance $σ^2$.

What does this mean? Is this not a variable anymore? Is this a function now? I see this in most books and such but I'm still unclear what exactly it means or what it does and etc.

In terms of regression, I know this variable is basically the random errors, but what does it mean if this vector of random errors follows a normal distribution?

I.I.D. means independent and identically distributed, so $epsilon$ is a vector of component random variables with the same distribution.

The meaning of "A follows an X distribution" is equivalent to saying that it "has a distribution," which is to say that it is a random quantity that can be determined only in probability.

In the example of regression that you refer to, $Y=f(X) + epsilon; epsilon stackreli.i.d.sim N(0,sigma^2)$, so the response variable $Y$ is equal to some function of the independent $X$ on average, and errors are normally distributed with mean zero, i.e. the observed $Y$ is not exactly $f(X)$.

A random variable $varepsilon sim mathrmN(0,sigma^2)$ is not the kind of variable considered when thinking about function arguments or solving equations, but actually represents the outcome of a random experiment. (Mathematically rigorously, but not so important, one would say: it is a function mapping from a sample space into the space in which the random variable lives.)

How can this be understood? A probability measure, like $mathrmN(0,sigma^2)$ assigns values to sets, so-called events. In this case, the probability of $varepsilon$ ending up in a set $A$ has probability
$$
mathrmN(0,sigma^2)(A) = int_A frac1sqrt2pisigma^2expleft(-frac12sigma^2 |x |^2 right) mathrmdx.
$$
That means, if you repeatedly saw i.i.d. (independent and identically distributed) $varepsilon$'s, they would (in the large data limit) on average end up in $A$, precisely $mathrmN(0,sigma^2)(A)cdot 100 %$ of the time.