explanation of correlation of stationary stochastic processes
You are mixing up two different notions. 

Your random process is a collection of random variables $\{X(t)\colon -\infty &lt; t &lt; \infty\}$, one random variable for each time instant. The autocorrelation function of the process is
$$R_X(t, \hat{t}) = E[X(t)X(\hat{t})], -\infty &lt; t, \hat{t} &lt; \infty$$
and is a two-dimensional function (meaning it has two arguments,
and if you try to visualize it, you should imagine a surface above 
(and possibly
below) the plane with coordinate axes $t$ and $\hat{t}$. Definitely above
because $R_X(t,t) = E[(X(t))^2]\geq 0$ for all $t$ (and processes for
which $R_X(t,t) = 0$ for all $t$ are kinda boring).

For a stationary process, $R(t,\hat{t})$ depends only on the
difference $\hat{t}-t$ of the two arguments, and it is common
practice to write it as a function of one variable $\tau = \hat{t}-t$.
Thus,
$$\mathcal R(\tau) = R(t,\hat{t}) = E[X(t)X(t+\tau)].$$

The other notion is that if the experiment is performed, then
each random variable takes on a numerical value. Thus, $X(t)$ takes
on a specific numerical value that we name as $x(t)$ while
$X(\hat{t})$ takes on some other numerical value that we name
as $x(\hat{t})$ and so on. Thus, we have a real-valued function
(or signal) $x(t)$ (nothing random about that!) for which the
autocorrelation function is defined as
$$R_x(\tau) = \int_{-\infty}^\infty x(t)x(t+\tau)\,\mathrm dt.$$
Note that $x(t)$ is called a sample path or realization
of the random process: it is what you observe on your oscilloscope
if you lift up your head from the math a little bit
and look at the real world.

It is every DSPer's fondest hope that $R_x(\tau)$ is the same function
as $\mathcal R(\tau)$. Unfortunately, nothing in what has been
said above justifies this hope. But, since people like to assume
that it is indeed true that $R_x(\tau) = \mathcal R_X(\tau)$, mathematicians have come
up with a notion called ergodicity which (for our purposes) says that
for an ergodic process, the ensemble means (expectations)
equal the corresponding time-averages.  As a practical matter,
all model-builders assume that the processes that they want
to model are indeed both stationary and ergodic,
and decide on what the mean and
the autocorrelation function etc should be by making observations of
actual sample paths etc.

For more than what you probably want to know about random processes,
see this answer of mine, and
for an example of a (boring) stationary random process that is 
not ergodic, see this other answer of mine.