How do I calculate the positions of objects in orbit? In a model of the solar system, given the Sun is at the point $(0, 0, 0)$, an given the six orbital elements for each object in orbit ($a$, $\epsilon$, $i$, $\Omega$, $\omega$, $M_0$), how can I calculate the relative position to the sun at a given time $t$?
You must solve the Kepler's equation:

$M = E - e \sin E$

(where $M$ is the mean anomaly, $e$ the eccentricity of the orbit, $E$ the eccentric anomaly).
As you can see, this is a transcendental function, so you will probably need to iterate through a calculator, in order to solve it.
Many of useful formulas to understand the parameters dependence are reported on the Wikipedia page as well. 

Here is another page with some examples. Just use the formulas and try to play with numbers, at some point you will realize at least orders of magnitude of the binary system elements.