I hope you already know most of that very condensed set of material. This is a long story. One needs to know what a smooth manifold is, differential geometry and Riemannian metrics, differential forms and homology and deRham cohomology theory.
Then the laplace operator associated to a metric is used to define harmonic forms and show under certain conditions how to represent the abstract deRham cohomology elements via harmonic forms, and hence decompose the deRham cohomology according to "type".
Then one does the whole subject over again in the complex case, complexifying the tangent and cotangent bundles, with hermitian metrics replacing riemannian ones, and with holomorphic forms contributing a small subclass of the harmonic forms in the decomposition of the cohomology. One obtains a fundamental invariant, the "canonical bundle", the complex line bundle constructed as the determinant of the holomorphic cotangent bundle.
The basic examples of these complex manifolds are smooth closed subsets of projective space, i.e. algebraic varieties, so one tries to isolate some special properties that those examples have, one of which is the "kahler" property, that the hermitian metric inherited from a projective embedding defines a natural (1,1) which is "closed". Not all Kahler manifolds in fact are projective but they do all have a good Hodge decomposition for their cohomology which makes them easier to study.
This is all basic machinery for studying complex manifolds. Other useful topics include sheaf cohomology, and the relation between line bundles and divisors, and criteria for existence of a projective embedding, i.e. when is a line bundle "very ample"? The relation with some earlier concepts is that a line bundle which gives an embedding has a cohomology (1,1) class which is essentially its kahler class, for the kahler structure inherited from the embedding.
Next one begins to look a examples. For dimension one, these are riemann surfaces, divided up by the genus, or more crudely into elliptic, parabolic, and hyperbolic types, according as g =0, g=1, or g >1. Equivalently, the canonical bundle and all its positive tensor powers has no ≠0 holomorphic sections, exactly one section (with no zeroes), or has some positive power which is actually very ample, i.e. has lots of sections. The hodge decomposition here just says that the space of deRham cohomology in degree one, is the direct sum of the spaces of holomorphic and antiholomorphic one forms. There are no holomorphic or antiholomorphic 2 forms, so the one dimensional deRham space in degree 2 is spanned by a single (1,1) form.There is no counterpart in dimension one of a true Calabi Yau manifold,a lthough the closest one is the torus of genus g=1. I.e. it shares with the Calabi Yau manifold the property that the canonical bundle is trivial, but it also has non zero one forms. Usually one considers as Calabi Yau only those more exotic manifolds with trivial canonical bundle and which are also without any ≠0 one forms, i.e. topologically speaking, those which are "simply connected".
These exotic objects occur first in complex dimension 2. . There is a simple formula for the canonical bundle of a hypersurface in projective n space, derived from the adjunction formula for how to restrict a meromorphic form to a hypersurface, i.e. how to restrict the canonical bundle of projective space to the hypersurface. The “tautological” line bundle on projective space, induced by the fact that the points of P^n are actually lines, is called O(-1) because its chern class is -1. The canonical bundle of P^n is O(-n-1) the n+1st tensor power of O(-1), Then for a smooth hypersurface of degree d in P^n, the canonical bundle is O(d-n-1). Thus for a plane cubic we get O(0) = O, the trivial canonical bundle. That is why a smooth plane cubic is a torus, i.e. is a Riemann surface of genus one.
Products of tori also have trivial canonical bundle as do other complex tori of form C^n/lattice. So these are not really exotic as Calabi Yau manifolds go. But we get some new ones from the adjunction formula as follows: every hypersurface in P^n is simply connected if n > 2, hence by the Hodge decomposition, also has no holomorphic one forms, (whereas complex tori have lots of one forms, also by the Hodge decomp if you like, since topologically they are products of circles). So we will get a new Calabi Yau surface in P^3 if we rig up the formula O(d-n-1) to come out zero. I.e. we want O(d-4) = O, so any smooth quartic surface in 3 space works.
Since they are simply connected there are no deRham one forms at all, and by Poincare duality also no 3 forms. In the hodge decomposition of forms according to type (p,q) neither p nor can exceed the complex dimension of the manifold, hence the top cohomology space, which is always one dimensional, i.e. the degree 4 deRham space is spanned by a single harmonic (2,2) form. This leaves only one interesting dimension, the deRham space of (closed mod exact) two forms. Since our surface has trivial canonical bundle, it does have a never zero holomorphic 2-form, whicht hen spans the Hodge space of all (2,0) forms. (Dividing any other holomorphic form by it would yield a global holomorphic function, necessarily constant on a compact complex manifold.) Hence the hodge space of (0,2) forms is also one dimensional. Since one can compute the space H^2 to have dimension 22, according to wikipedia, this gives us the whole “hodge diamond”. I.e. the interesting middle row is [ 1 20 1], the top and bottom vertices are 1’s, and the other two rows are zeroes.
These are special examples of K-3 surfaces, the 2 dimensional analog of Calabi Yau’s. They are really interesting and the subject of many papers. E.g. for most curves of genus g > 2 not only is some power of their canonical bundle very ample, but the canonical bundle itself is very ample. A curve embedded in projective by its canonical bundle, i.e. such that a hyperplane cuts out the zeroes of a holomorphic one form on the curve, is called a canonical curve. Hence a smooth plane quartic is a canonical curve. Now on a quartic surface in space, every hyperplane cuts out a plane quartic, hence a canonical curve. This property of K-3 surfaces that hyperplane sections are canonical curves generalizes, and allows one to define the genus of embedded K-3 surface as the genus of this canonical curve. (I am a little shaky on this.)
There are a few other surfaces that have similar properties to K-3 surfaces, such as the quotient of a K-3 surface by an involution, or an ”Enriques” surface. Also one can get another surface with similar numerical properties by taking a finite quotient of a 2-torus.
There are infinitely many families of algebraic K-3 surfaces, but interestingly they all fit together into one connected family if we fill in the holes between them by allowing non algebraic kahler examples. This is one good motivation for looking at Kahler surfaces that may not be algebraic. I.e. there is one analytic family of kahler K-3 surfaces, on which there are infinitely many distinct families of projective algebraic ones. (I am on shaky ground here too.)
Ok, now we go up to three dimensions, and can define a Calabi Yau 3fold as a complex kahler 3 fold with trivial canonical bundle, I,e, having a holomorphic 3 form with no zeroes, and perhaps also having no ≠0 one forms, to rule out 3-tori. Then analogously to the case of surfaces, we can produce these by looking for smooth hypersurfaces of degree d in P^n that make the canonical bundle O(d-n-1) come out trivial. I.e., we want d = n+1, but we also want n = 4, so that a hypersurface is a threefold, so we get d = 5, and we want a smooth quintic in P^4.
Now it is true as stated in the notes linked above, that the simplest case is the fermat quintic with equation a sum of fifth powers of all the variables, but I believe that case is too special for some purposes. I.e. a more general example has only a finite number of complex projective lines on it (2875?), and these lines are important for string theory (maybe the lines are among the "strings"), whereas the fermat example I believe has atypically an infinite set of such lines. I am also on shaky ground here.
If you are a beginner, as I am, I might suggest you take a look at K-3 surfaces, and quintic threefolds to start. I hope this is helpful. I.e. I have grown tired in my old age of reading notes that detail endlessly an enormous formalism without any explanation of what it means. So i have tried to put this subject in some context.
