Basic: Why are Slope, Intercept in Regression considered Random Variables? Sorry if this is too basic.
In an OLS regression given by
$y=ax+b$
$b$ intercept, $a$ the slope.
Then $a,b$ are not numbers but random variables. 
I find this confusing since I start with data points $(x_1,y_1),(x_2,y_2),..(x_n,y_n) ; x_i \neq x_j $ when $i \neq j $. Then we find
 the line of best fit, which would give us actual numbers $a,b$, so that these seem to be constants, not (Random) variables. 
Is the reason these are called variables that I am selecting just one
  of many possible values $y_j$ for a given variable $x_j$?    
 A: Estimators of slope and intercept are random variables because they're functions of the responses, which are random variables.
New samples would lead to different estimates (because - even assuming fixed $x$'s - you'd have different realizations of each of the n corresponding sets of $Y$'s)
If we set the situation up to make the variables and their realizations a little more distinct, the situation may become clearer; taking the $x$'s as fixed (for simplicity of exposition), you have $$Y_i = ax_i + b + \epsilon_i$$ where $\epsilon_i$ is the error term. You draw a sample with that set of $x$'s and you observe a corresponding set of $y$'s, corresponding to a particular realization of the $\epsilon_i$. Let's call that set of observed $y$ values $\mathbf{y}^{(1)}$. We repeat our sampling procedure at the same set of x-values and obtain a new set of responses, $\mathbf{y}^{(2)}$ and we keep going, up to $\mathbf{y}^{(k)}$ say. Each realization will have its own slope and intercept, so realization $j$ has $a^{(j)}$ and intercept $b^{(j)}$, which will be a function of $\mathbf{y}^{(j)}$. 
