What is the mathematical name of this kind of rotating ellipse? 
Is there a general name in mathematics and/or physics applied to the kind of shape/motion shown in the attached picture? I need a word to refer to this shape/motion.
 A: Informally these are called 'spirograph' curves but there is also a mathematical name for these:  hypotrochoids.  These are parametric curves $$f : \mathbb R \to \mathbb R^2$$ of the form
$$\begin{align}
x(\theta) &= (R - r) \cos \theta + d \cos \left(\frac{R - r}{R} \theta\right), \\
y(\theta) &= (R - r) \sin \theta - d \sin \left(\frac{R - r}{R} \theta\right), \\
\theta & \in \left[0, \frac{2 \pi \operatorname{lcm}(R,r)}{R}\right).
\end{align}$$
Here, $R$ and $r$ are radii, and $d$ is a distance parameter.  Your specific curve can be generated for the choices $R = 7$, $r = 3$, $d = 6$:

Many other choices are possible, each generating various interesting curves and patterns.
A: Locus generated by means of rolling between two (or more) objects (without slipping) is called roulette and whereas locus by means of sliding between objects is called glissette.
In your sketch, the curve can be possibly generated by a spirograph.  You may say your curve is a kind of hypotrochoid which is determined by three parameters.
