(There is also a
german version of this page.)
This nice animation of Poncelet's theorem
is due to
W. Barth.
What is it about?
Poncelet's theorem is a famous
porism

in fact it may be viewed as a
»prototype«
of all such theorems.
It is about two conics in the (projective) plane that do not
touch each other

in the animation these are the two circles.
Here is the statement of the theorem:
If there is an ngon that is
inscribed
into one of the conics and
circumscribed
on the other conic,
then there are
infinitely many
such ngons.
More precisely, there is then in fact
such an ngon through any point
on the conics.
What has the theorem got do to with the animation?
Well, it is Poncelet's theorem which makes
such an animation possible in the first place.
This is because in practice the theorem means that an ngon
that is inscribed/circumscribed between two conics may be
»continuously deformed«.
You can see this in the pentagon that is inscribed into the
outer conic and circumscribed on the inner conic:
Poncelet's theorem guarantees that we may
»rotate«
the pentagon without breaking it apart.
Where can I learn more
about Poncelet's theorem?
Griffiths and Harris have given a modern proof of Poncelet's
theorem in their paper

Ph. Griffiths, J. Harris:
A Poncelet theorem in space.
Comment. Math. Helvetici 52, 145160 (1977)
The proof takes place in
algebraic geometry, and it
reveals that an
elliptic curve
is at work behind the scenes here.
A quick internet search provides interesting sites, too  for
instance the nice
Staatsexamensarbeit
by Ulrike Herr.
If you now got interested in further theorems of this kind,
you will find a collection of them in the paper
along with modern proofs and hints to the literature.
(Also, you learn there that almost always an elliptic curve is
working behind the scenes  it is responsible
for the porism to work.)