Similar to gowers's answer about group actions, a module over a ring R is an abelian group M together with a function $f:R\times M \to M$ that satisfies certain properties. It may set the beginner's mind at ease to hear, "They're just like vector spaces except over arbitrary rings instead of only fields," which is misleading in itself but is a good mnemonic for remembering the definition. However, I usually find it more intuitive to think of a module over R as a homomorphism from R to the endomorphism ring of an abelian group, and with this definition no mnemonic is necessary.
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