TITLE: Down-to-earth expositions of Hodge theory
QUESTION [36 upvotes]: What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?
Of course, since I haven't found a (for me) readable introduction, I don't know what I should learn about Hodge theory, but at least I'm looking for:

reasons to expect that solutions to the laplacian (rather than other operators) should be studied
examples where the harmonic forms can be explicitly written down
how to think of harmonic forms in general
the hodge diamonds (and how to compute them!) of the standard examples of projective varieties

So far I have read (parts of) the relevant parts of Griffiths-Harris and lecture notes on the web, but still don't how understand how to do computations with Hodge theory, or for which varieties I should expect to be able to use it.

REPLY [15 votes]: I thought I'd add a few words to answer the questions "reasons to expect that solutions to the laplacian/how to think of harmonic forms". I tend to think of the Hodge theorem as an infinite dimensional version of the  least squares. De Rham's theorem tells us that the cohomology of a manifold can be identified with the space of closed forms modulo exact forms. But this raises the question how do we choose a good a representative for cohomology class? Suppose our manifold is compact and orientable, and that we have chosen a Riemanninan metric which allows us to measure size. Then the natural representative would the one  that minimizes the norm. This is precisely the harmonic representative. Here is the explanation.
If $\Delta\alpha=0$, where $\Delta = dd^*+d^*d$, then $$||d\alpha||^2+||d^\ast\alpha||^2=\langle\Delta\alpha,\alpha\rangle=0$$
so $\alpha$ is closed and co-closed (and conversely such forms are harmonic). Therefore $\alpha$ determines a cohomology class. Any other form in the class is given by $\alpha+d\beta$. We have
$$||\alpha+d\beta||^2= ||\alpha||^2+||d\beta||^2+\langle d^*\alpha,\beta\rangle=
||\alpha||^2+||d\beta||^2\ge ||\alpha||^2$$
So indeed, $\alpha$ has the smallest norm in its class.
For applications to algebraic geometry one needs the harmonic theory to play well with the complex geometry (e.g. holomorphic forms should be harmonic). For this, we need a Kähler metric as described in Alex Degtyarev's answer.