I have seen the thin lens equation, which describes the following relation.
$$\frac{1}{\text{image distance}} + \frac{1}{\text{object distance}} = \frac{1}{\text{focal length}} \tag{1}$$
However, I've also seen the following relation.
$$\frac{\text{object height}}{\text{image height}} = \frac{\text{object distance}}{\text{focal length}} \tag{2}$$
I was wondering how you get to one from the other, and what exactly the difference is, in terms of what they're trying to model?
A correct formula, of which #2 might be an incorrect interpretation, would be
(object height)/(image height) = (object distance)/(focal distance)
where focal distance is the distance from the lens to the image.
What's different between Formula 1 and the correct Formula 2 is that Formula 1 calculates the image position from the object position, given the focal length of the lens; while correct Formula 2 calculates the magnification, given the object and image positions. The two (#1 and correct #2) can be combined in various ways to yield, e.g., the necessary focal length and lens position to obtain a given magnification with object and image at given positions.
You don't. They are derived independently, from the geometry of the ray tracing figure. The first one is a relationship between the positions of the image and object. The second one allows to calculate the magnification of the lens. You can use the first one to replace one of the distances in the second formula so you can get the magnification as a function of the position of the object (for example), for a given focal length.
