The Yoneda Lemma basically says that if you take your favorite object, and just consider morphisms to (or from) that object, that tells you everything you need to know.<p>So for example, if I'm in the category of groups, and I look at Z/3, then what are morphisms from Z/3 to a given group G? Well those are just the 3-torsion elements of G (elements such that g^3 = identity). That is, the image of 1 (mod 3) must be such an element, and conversely such an element determines a morphism by sending 1 (mod 3) there.<p>Yoneda says this actually characterizes the group Z/3. The language used is that Z/3 "represents" the functor taking a group to the set of its 3-torsion elements.<p>This can be useful when the object you're trying to characterize is more complicated than the functor it represents.<p>In algebraic number theory, the functor taking a field k to the set of pairs (E,p) where E is an elliptic curve and p is a point of order 593229 defined over k, <i>is representable</i> by some equation, but the equation would be opaque and maybe not so useful.