TITLE: Can someone explain ideals to me?
QUESTION [23 upvotes]: I'm struggling with the idea of ideals (both the definitions and the common notation).
I'm in a basic collegiate algebra course, just looking for a bit of help.
As simply defined as possible, if you could.
thanks!

REPLY [4 votes]: One characterisation that is implicit in some of the other answers, especially Sasha's, is the following: 

Ideals are exactly those subsets which can arise as the kernel of a ring homomorphism.

That kernels are ideals is immediate (and can hence motivate the definition of an ideal), and the quotient construction (modding out) yields an epimorphism whose kernel is a given ideal.