What is the difference between constant elasticity of substitution and elasticity of substitution? What is the difference between constant elasticity of substitution and elasticity of substitution? Are these formulas related? How are they different concepts? I just did a whole problem set involving these and got the right answers. But my thought process was all math. I don't feel like I understand them economically. And that's what I want explained.

By CES, I mean
$$U(x,y) = A \left(\alpha x^{-\rho} + (1-\alpha) y^{-\rho}\right)^{-\frac{1}{\rho}}$$

By ES, I mean 

$$\sigma = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U_y}{U_x}\right)}$$
Well, you didn't get the math all right, otherwise you might have seen it yourself! 

By Elasticity of substitution, you should refer to

$$ \sigma(x,y) = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U_y}{U_x}\right)} $$

Then, constant elasticity of substitution is a property of a function $U$ such that for some constant $\bar \sigma$,

$$ \sigma(x,y) = \bar\sigma \, \forall y, x$$

That is, the elasticity doesn't depend on $x,y$. What you showed as $U$ is a function that has exactly that feature. If you compute $\sigma(x,y)$ (do it!), you will see that

$$ \sigma(x,y) = \bar\sigma = \frac{1}{1+\rho}$$

That is, the elasticity of your $U$ function is constant for any allocation. Hence, we refer to that class of function (for different $\rho$) to Constant-Elasticity-of-Substitution-Functions (with share parameters $\alpha$). Cobb-Douglas preferences would be a special (limiting) case of that, with $\sigma(x,y) = 1$

Example

Say, we produce happiness ($U$) with apples ($x$) and oranges ($y$). Lets look at the allocation $\{5,1\}$. We only have one orange, but we would really want more. Hence, we are willing to give up 2 apples to get one orange, and we would be indifferent:

$$ U(5,1) = U(3,2)$$

It is very easy to substitute apples with oranges, we have a high elasticity of substitution. 

Now look at the allocation $\{1,6\}$. Do you think that you would again be willing to lose $40\%$ of $x$ to get an increase of $50\%$ in $y$? Is it true that

$$ U(1,6) = U(0.6, 12)$$

If yes, that means that the elasticity of substitution of $U$ (or my rather handwaving approximation to it) is the same at the two allocations $\{1,6\}$ and $\{5,1\}$. If that was true for all possible allocations, $U$ was CES.

There are some reasons why you would think some preferences are perhaps not CES: For example, if $x$ and $y$ are good complements. Then, when you have little of $x$, you're willing to give up a lot for the other, but that elasticity might decrease the closer you are to equality between the two inputs.