What is meant by "vacuum polarization"?

In quantum field theory particles like electrons are not the fundamental objects. Instead the fundamental objects are the quantum fields. These fields have quantum states, just like all quantum objects, and some of these states appear to us as particles. So when we observe an electron we are actually observing a particular state of the quantum field that looks to us like an electron.

Like all quantum objects the quantum fields have a ground state i.e. a lowest energy state. If a field was completely non-interacting, i.e. if the particles the field described did not interact with each other or with anything else, then the field would be zero in the ground state and would have zero energy. However the particles in the Standard Model all interact with each other and as a result all the fields interact and become entangled, and the result is that the ground state is hideously complicated and far too complicated for us to calculate it. Instead we have to approximate its properties using perturbation theory, and this leads us to the popular science description of the vacuum as a roiling mass of virtual particles (though this is a metaphor and not actually true).

So far so good, but if we introduce an electron then we no longer have a vacuum and the vacuum near the electron changes as a result of the perturbation introduced by the presence of the electron. In popular science terms the electron interacts with all the virtual particles present in the vacuum state, and this slightly changes the properties of the electron. In the case of the Lamb shift we have an electron bound into a hydrogen atom, and the interaction with the vacuum slightly changes the energy of the hydrogen atom. It is a very small change, but it is measurable because the shifts in the energy levels of the hydrogen atom can be detected in its absorption and emission spectra.

So the vacuum polarisation is the change in the quantum field ground state caused by the presence of a charged particle like an electron. If you look at the Wikipedia article you linked it says:

In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic field.

Though we should emphasise that the virtual electron–positron pairs are just a computational device for calculating the quantum field ground state.