What does the Cartesian equation of a Parametric function tell us? I've been taught that a Parametric function can be converted into a Cartesian one by eliminating the parameter $t$ but I've never been taught of how it specifically relates to the Parametric. Does it form an exact replica or what?
The parametric curve converted into a cartesian curve, when it is possible to do so, passes through all of the same points as the original. In that sense, it is an "exact replica." The information lost is how fast the curve gets traced out. Depending on the situation, this piece of information may or may not be important.
parametric equn of parabola

$$ x= 2 f t ;\, y = f t^2 $$

Find $t$ from first part

$$  t = x/(2 f) $$

Plug it into second part

$$ y = f (x/(2f) ) ^2 $$
simplify

$$ 4f y = x ^2 $$
Both ways it represents the same curve.

If $t$ represents time we know how the speed of x,y components are individually 
represented. Here x speed is constant but y speed increase with time.