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.
