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

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. 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. 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. This can be useful when the object you're trying to characterize is more complicated than the functor it represents. 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, is representable by some equation, but the equation would be opaque and maybe not so useful.