Assumption of a Random error term in a regression In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then proceeded to explain that this error term is normally distributed and has a mean zero. 
The error term is what is confusing me. What exactly does random mean? My back ground in statistics is very low level, but I understand that a random variable is defined as a mapping from a sample space to the real numbers. This definition makes sense, but the assumption of a zero mean is what I get tripped up on. How can we assume this fact? 
I've been trying to think about it intuitively, and can only think that in regards to the real numbers, zero is in a sense "the middle ground" and splits up the reals into 2 "equal length" parts. However, I know that the reals are uncountable so this may be a case where my intuition is incorrect. 
I apologize in advance if my question is confusing.

The assumption of mean 0 is a normalization that must be made because you already have a constant term in the regression. It relates to the issue of identification - that you as the researcher cannot tell the difference between the constant term in the regression and the mean of the error term. 
Proof: Suppose that $\epsilon$ is not mean 0
Let $\bar{\epsilon}$ denote the mean of $\epsilon$. Then I can re-write your model as 
$Y = (\alpha + \bar{\epsilon}) + \beta X + (\epsilon - \bar{\epsilon})$. 
let $\tilde{\alpha} = \alpha + \bar{\epsilon} $ and $\tilde{\epsilon} = \alpha + \bar{\epsilon}$
-->$Y = \tilde{\alpha}+ \beta X + \tilde{\epsilon} $. 
This model is identical to yours except it now has a mean-zero error term and the new constant combines the old constant and the mean of the original error term.
