Vacuum polarisation in QED - why is it significant to renormalisation? I have followed along for the derivation of the amplitude of the 2-photon vacuum polarisation and the book says the result is important for the renormalisation of QED, why is this?
 A: A generic connected Feynman diagram can always be decomposed into a string of one-particle irreducible (1PI) diagrams connected together by propagators. The reasoning is very simple: a Feynman diagram is said to be 1PI when it cannot be divided in two meanigful Feynman diagrams by cutting one internal line in two. A Feynman diagram is either 1PI or not. If it's not then you can cut it in two by splitting one internal propagator. You can repeat the question to the two diagrams you get in this way. The point is that you will continue splitting the diagrams until you reach 1PI diagrams. In the end your original diagram which was not 1PI was seem to be a string of 1PI diagrams connected together by propagators.
Now what this means is that if your 1PI diagrams do not diverge, no diagram will, since all the diagrams are built out of the 1PI diagrams just including propagators. This means that if you want to remove the divergences of the Feynman diagrams you must focus on the 1PI diagrams.
Now, it turns out that "vacuum polarisation" in QED is just the 1PI contributions to the photon propagator. In that case, when you study vacuum polarization you are actually making sure that one of the 1PI diagrams is well-defined.
A: Briefly speaking, the renomalization of QED is structured as follows:

*

*Any diagram can be built from 1-particle irreducible (1PI) diagrams. In QED in 4D one may show that there are only 7 superficially divergent 1PI diagrams without  divergent subdiagrams, cf. e.g. Ref. 1.


*Excluding 1 vacuum diagram, 2 that vanish by Furry's theorem, and 1 finite, there remain actually only 3 divergent 1PI diagrams: The self-energy/vacuum polarizations of the photon and electron fields, and their 3-vertex diagram. OP asks about 1 of these 3.


*4 $Z$-factors need to be renormalized: The wavefunction renormalizations of the photon and electron fields, the $Z$-factor for their coupling constant $e$, and the $Z$-factor for the electron mass, cf. e.g. Ref. 2.
References:

*

*M.E. Peskin & D.V. Schroeder, An Intro to QFT; Section 10.1.


*M. Srednicki, QFT, 2007; Chapter 62. A prepublication draft PDF file is available here.
A: Very very briefly speaking: Vacuum polarization refers to the process by which a point particle like an electron can interact with electron/positron pairs that pop briefly out of the vacuum nearby. When that occurs, the pairs orient themselves with respect to the "real" electron so the positrons (+ charge) face that electron and the electrons point away from it- for an instant.
Over time, this makes the otherwise empty vacuum behave a tiny bit like a dielectric containing charge dipoles that can respond to electric fields, and if observed from far away, the electron's charge appears to have been partially screened by this  "vacuum dielectric" effect.
The process of renormalization then involves taking that screening effect explicitly into account when calculating the quantum mechanical behavior of what we thought of as a "bare" electron, but which is actually "dressed up" by what can be thought of as a cloud of  positron/electron pairs that surround it.
This is the connection between vacuum polarization and renormalization- as I crudely understand it i.e., without the use of Feynman diagrams.
A: Renormalization is the name we give to the following idea.
In order to calculate what quantum fields do, we have to perform integrals. We need a mathematical method to make these integrals tractable. One method is to start with states of the non-interacting fields, and construct Feynman diagrams, and integrate. However some of these integrals diverge. On investigation, it turns out that one way to see why we got a diverging integral is to realise that we tried to construct our theory out of hypothetical physical states (the free particle or bare states) that are infinitely far-removed from the actual physical states (states of the interacting fields). It is a bit like doing the integral
$$
\int_a^b \frac{1}{x} dx = \ln(a/b)
$$
by trying to adopt the expression
$$
\int_a^b f(x) dx = \int_a^0 f(x) dx + \int_0^b f(x) dx
$$
which seems ok at first but breaks down when $f(x) = 1/x$. In this integral one is trying to write a perfectly well-defined finite integral as the difference between two divergent integrals, which is not a well-defined idea. So to avoid that we can instead adopt
$$
\int_a^b f(x) dx = \lim_{\epsilon \rightarrow 0} 
\left[\int_a^\epsilon f(x) dx + \int_\epsilon^b f(x) dx \right]
$$
and now everything is well-defined.
The bare states of electrons and photons---the ones seen in a theory of a free field of either with no interaction term---are serving the role of $\epsilon$ in the above. They are infinitely far removed from the actual physical states, in the sense that their energy differs by an infinite amount from the energy actually there in the physical fields. But if we are careful enough then we can set the calculation up to get the difference between the two integrals (which taken one at a time would diverge) and thus get the difference between the energies of two physical states, or, more generally, some other quantity, such as a quantum amplitude to evolve from one physical state to another.
Finally, then, what has all this got to do with vacuum polarization? Other answers have already mentioned how we just get a small number of Feynman diagrams we need to renormalize, and one of these is the one asked about. The name 'vacuum polarization' is a name for the state of the physical fields in this scenario. A physical photon is not the same thing as a bare photon, and a physical electron is not the same thing as a bare electron. We can imagine the physical electron as if it were a bare electron surrounded by a cloud of bare virtual electron-positron pairs which are arranged as dipoles giving a net polarization. But this picture is just giving us a way to think about physical electrons in terms of bare electrons and positrons, and bare electrons and positrons are figments of our imagination: they are calculational tools, states of a field (the non-interacting field) which does not exist. Because the actual fields do interact.
It has long been recognized that it ought to be possible to do all this maths without the use of renormaliztion at all (e.g. by working with physical states throughout and not invoking bare states), but the method has been worked out carefully in a rich array of circumstances and it remains the best one we have for many purposes.
