Where can I read about elliptic operators on manifolds?

In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators:

Let $P : E → F \ $ be an elliptic differential operator on a compact manifold. Assume that $E$ and $F$ are of the same rank, and are equipped with metrics. Then $\operatorname{ker} P \subset C^{\infty}(E)\ $ is finite-dimensional, $P( C^{\infty}(E)) \subset C^{\infty}(F) \ $ is closed and of finite codimension, and we have a decomposition as an orthogonal direct sum (for the $L^2$ metric) $$ C^{\infty}(E) = \operatorname{ker} P \oplus P^{\ast}( C^{\infty}(F)), $$ which she then applies to the Laplace operator.

Unfortunately, her reference for this (Demailly - Théorie de Hodge L2 et Théor`emes d'annulation) is in French, which I don't understand.

I am primarily hoping for a reference on elliptic operators which proves this theorem; however, at the same time I also feel like I should know more about elliptic operators in general, and 'how they can be used in geometry' (can they, actually? I'm thinking about Atiyah-Singer here, which I don't understand at all, so I'm not sure.), so if you can give me a reference about that, it would be much appreciated, too.

One more suggestion: Wells Differential Geometry on Complex Manifolds has a complete proof, and focuses on the sort of examples which will interest someone who is reading Voisin's book.

The book Topology and Analysis: Atiyah-Singer Index Formula and Gauge-theoretic Physics by Booss and Bleecker (Springer , Universitxt) seems to be exactly what you are loooking for.
It is an introduction to the Atiyah-Singer index formula, with all prerequisites carefully developed :Fredholm operators, (pseudo-)differential operators on manifolds, Sobolev spaces, vector bundles and much more. The style is quite friendly, with many examples,remarks, exercises, interspersed throughout the text.
To sum up, I find the blend of analysis, manifold theory and topology in this book quite remarkable and unusual.

I am only really familiar with the German original version, written by Booss alone. The English translation I refer to is enriched by about 100 pages due to Bleecker and devoted to applications in mathematical physics: they seem very interesting but I haven't looked at them seriously .