You might try imagining what the equation is saying in geometric terms. (Imagine complex numbers as points in the complex plane.)
To do that, notice that $|z-4i|$ is just the distance from $z$ to the point $4i$ in the complex plane. Similarly, $|z+4i|=|z-(-4i)|$ is the distance from $z$ to $-4i$.
The equation is saying that the sum of these two distances is equal to $10$ for all $z$ in your curve. But this is the familiar definition of the ellipse, quoting Wikipedia: an ellipse is "the set of points such that the sum of the distances to two fixed points (the foci) is constant". In our case the foci are $4i$ and $-4i$ and the constant is $C = 10$.
Note that from these numbers you may also calculate (using the usual formulas from elementary geometry, for example the ones given on Wikipedia*) the major ($a=5$) and minor ($b=3$) axes of the ellipse, so this gives you all the information needed to completely describe it.
(*): in our case, we know the distance from the center of the ellipse to each of the foci, it is given by $f=4$. We also know the constant sum of the distances from the foci, $C=10$. Now, simply use the formulas $f^2 = a^2 - b^2$ and $2a=C$ and the problem is solved.