Question: I asked 3 professors at my university and none gave me a clear cut answer, but instead merely told me qualities of this notation.
Here is what I understand so far from what they told me:   

1)Treat the top variable as as variable when finding the derivative  

2)Treat the bottom variable as a constant when finding derivative  

3)It it said "Find x with respect to y", but what exactly does that mean? What does it mean for something to be in respect to something else?

It seems like $\frac{dx}{dy}$ notation changes according to values in the problem. For instance, If $y = x^3 + 2x$ and $\frac{dx}{dt} = 5$, find $\frac{dy}{dt}$ when $x=2$. Why do the values of $\frac{dx}{dy}$ change in this problem and how do I solve this?

Answer: 
The existing answers are great, but I thought I could add to the question. Sometimes we use $\frac{dx}{dy}$ in (differential) equations as quantities. As in $\frac{dx}{dy}$ represents "$x$ having been differentiated with respect to $y$." It's like saying $+$ is a defined operator (known as adding or addition), that can operate on $a$ and $b$. So the operation is $+$ and we can write $a+b$ to be a *quantity*; namely, the quantity of $a$ having been added to $b$.

Similarly, $\frac{d}{dy}$ is an operator. $\frac{d}{dy}(x)$ means to perform the operation of $\frac{d}{dy}$ onto $x$. **Specifically, $\frac{d}{dy}(x)$ means "differentiate $x$ with respect to the variable $y$."**

So in the end, **$\frac{dx}{dy}$ represents the quantity of $x$ having been differentiated with respect to $y$.** This is why textbooks often say "find $\frac{dx}{dy}$", because we wish to find what $x$ is upon differentiating it with respect to the variable $y$.

I see it as: $\frac{dx}{dy}$ is the end result of $x$ that underwent differentiation, and $\frac{d}{dy}$ is the actual operator, much like how $+$ is an operator.
Here is an example:

Let $y=x^2.$ Then you want to find the derivative of $y$ with respect to $x$. Which means, you want to find $\frac{dy}{dx}$. In another form, $\frac{d}{dx}(x^2)$. Since $y=x^2$, then $\frac{d}{dx}(x^2)=\frac{d}{dx}(y)=\frac{dy}{dx}.$ Its a simplification of notation in a general sense.