Very briefly ...
==== Start RSA recap
Given an integer n>1, the numbers a s.t. 0<=a<n and gcd(a,n) form a group under multiplication. That means that for every e with gcd(e,n)=1 there is a d s.t. d.e=1 (mod n).
Now take n=pq where p and q are primes. The function phi(n) counts how many elements are co-prime to n, and since n=pq that turns out to be (p-1)(q-1). So phi(n)=(p-1)(q-1). I'm going to write r=phi(n).
Take any e with gcd(e,r)=1. We can compute d s.t. d.e=1 (mod r), which means d.e = k.r+1 for some k. (Note: I'm doing this mod r, not mod n.)
Right.
Now take a message M (with 0<=M<n and gcd(M,n)=1) and compute E=M^e. We can do that fairly quickly using an adapted Russian Peasant Multiplication algorithm. This number "looks random" in some sense. You can transmit it to someone else.
They compute D=E^d. So what's that? Well, working modulo n:
D = E^d
= (M^e)^d
= M^(d.e)
= M^(k.r+1)
= M^(k.r) x M
= (M^r)^k x M
Therefore we can recover M, so we can decrypt E.
So if you publish n and e, but keep d secret, people can send you E=M^e (mod n) and only you can read it.
Probably.
If someone can compute phi(n) then they can compute d from e and n, but we think that's the same as factoring n. Similarly, if you can compute discrete logarithms, but that seems to be about as hard as factoring.
==== End RSA recap.
All of this can be cast more abstractly in the group (Z/nZ, * ). Doing so gives us the same system in more generality. This is what ECC does. You choose and publish an elliptic curve - C. Then you choose an element, e, and compute its inverse d in C. You encrypt a message M by taking e.M (remembering that in ECC we usually use + as the operation symbol instead of * - so this is the equivalent of M^e).
And it all works.
Possibly someone who knows more about this than I will find gaping holes in the above, but I think that should get you started.
To address the other question:
> I always thought that elliptic curves were
> an algorithm to break cryptography like RSA
Both RSA and ECC are based on the idea that exponentials are easy to compute, and undoing them is hard. In the case of RSA, you're exponentiating in Z/nZ and in ECC you're exponentiating in the group of points that arises from the chosen elliptic curve.
The first book in that list is a good text. It includes an appendix that includes almost everything you need to know about projective geometry to understand the theory of elliptic curves.