What is an elliptic curve in simple terms?

What is the elliptic curve method?

I've seen some interesting articles on the web about how elliptic curve is used to encrypt passwords.

I just wanted to clarify ? Is it like we can calculate the elliptic curve parameters based on the password and so on? Also how do you calculate P,G,Y and x0? I was a bit skeptical at first that this had anything to do with encryption of any kind, but I figured I would at least give it a try. In your example, I have left x0 and P as numbers. G is not used in this scheme because it can be found easily using a discrete logarithm. P and x0 are also not used because they are already provided as part of a standard library. However, y might be used, but it must be a small integer that the user knows ahead of time. (It doesn't have to be unique, but for security this would best.)
My own version does use the curves from NIST standard #6 v2.3 (which is defined in section 6.

The key is to take the two groups in the ratio Y:Y(G). There are two groups here that are known, and they can be found with elliptic curve exponentiation and modular multiplicative inverse. The reason the groups are in that ratio is for efficiency: I can take only two points to find the other elements of the curve. If the points were not in this ratio, a different point could be selected to get the second point.

What is the formula for the elliptic curve?

Is there any theory that explains why the elliptic curves are important, so important for cryptology etc.

The two points of interest from my perspective might be: How can such an unimpressive looking equation arise from a physical model (in the case of elliptic curve cryptography it's the elliptic curve point-addition, but it doesn't have to be). The relation between elliptic curves and number theory and algebra. For 1, it's really a question of how exactly the mathematics behind elliptic curve cryptography arose. When I was your age I was more interested in how things work; a simple case of a physical example would be a hammer (in which we learn the difference between static properties, which have always existed, and kinetic properties, which appear when a force is applied to a mass). An interesting case of a physical example is the way that the force of gravity is a property of space itself, that the gravitational force on matter does not change because its mass doesn't change. You need some understanding of conservation of energy.

For 2, I think it would be worth reading about Fermat's Last Theorem or other statements of the Pythagorean theorem, where geometry enters directly as the statement is being proved. If you read about these ideas you'll see that it makes sense for geometry and algebra to mix like this; a good exercise is for me to list a few examples of when an approach taken to solving a problem in one field leads to a new mathematical formulation.