Please write it as simple as possible.
migrated from math.stackexchange.com Feb 28 '12 at 7:30
(I only know the importance, not the mathematical treatment &c, but I doubt you want that)
Noether's theorem let's us obtain conservation laws. Conservation laws are pretty much the life of physics. If you want to calculate the outcome of any process, you have to see what is conserved in the process. Without these laws, you'd be left with an incomplete system. Mathematically, you'd have more variables than equations.
Noether's theorem let's us obtain conservation laws in a beautiful way: Any symmetry in the system gives rise to a conservation law/invariant. This is somewhat semi-intuitive, so I'll give an example (lifted from The Theory of Almost Everything by Robert Oerter):
Imagine a skateboarder in a half-pipe. He can skateboard in two ways: along the direction of the pipe, and perpendicular to it (in an oscillatory fashion). Now, let us look at what happens when we shift the pipe. If nothing apparently changes on such a shift ("space shift"), then we have a conservation law. So lets say we shift it along the direction of the pipe. Obviously, the skateboarder will not feel a thing. But, if we shift the pipe in the direction perpendicular to its length, the skateboarder will be floating in the air, and he definitely notices a change. This is shown in the below diagram. The black arrow indicates how I shifted the pipe. The red arrow shows the (relative) shift of the skateboarder, and the red stickfigure is the final (relative) position of the skateboarder.
So, what is the conserved quantity? It should be conserved along the pipe but not perpendicular to it. Over here, it is the velocity (actually momentum). The skateboarder will have constant velocity along the pipe, but his velocity can change in the perpendicular direction.
It seems more obvious how the theorem works now; if there is a type of symmetry, then doing a "shift" according to the symmetry does not change anything, and thus something is invariant or conserved.
More about it
In modern physics, you see symmetries left right and center. A lot of focus is put on symmetries, as they let you obtain conservation laws without the need of specifying the law as an axiom.
A few more symmetries and their conservation laws:
Note that QCD color is conserved in a trivial way in scattering--- everything that comes in comes out neutral. This conservation law is only a short-distance thing.
Noether theorem is one of the critical parts of the modern physics which puts a foundation to the connection between symmetry and conservation laws.
I would say that it is extremely important, but its importance is not too well-known by the students. It was even missing from some old theoretical physics courses. Probably, due to the fact that in most simple cases the statement of this theorem is obvious.