As for **category theory**, I don't think that there is a motivating example which has not already a category theoretic flavor. The leading theme is to unify and then generalize constructions resp. arguments, which come up in all areas of mathematics. Historically, natural transformations were introduced for the foundations of homology theory of topological spaces. But to start with an easy example, you may observe that for abelian groups $A,B,C$ there is a canonical isomorphism $(A \oplus B) \oplus C \cong A \oplus (B \oplus C)$, which reminds you of other *associativity* results such as $(X \cup Y) \cup Z \cong X \cup (Y \cup Z)$ for sets (here $\cup$ means disjoint union). Within category theory, you can see what's the real content of this: direct sum and disjoint unions are examples of coproducts, and coproducts are always associative. Even more striking, Yoneda's Lemma, which lies at the heart of foundations of category theory, tells you that the case of sets already settles the general case!

But category theory is more than just a language, it also provides general constructions: Assume you want to *approximate* a theory with another theory. This may be formalized by finding an adjunction between two categories. Freyds/Special Adjoint Functor Theorem tell you when this is possible. Although in some situations you can't write down the adjunction, the only thing you need is to know that it exists. For example, what is the categorical coproduct of an infinite family of compact hausdorff spaces? Can you write it down without using Stone-Cech?

There are also somewhat *global* motivations: Some Theories behave like other theories, and thus you may develope a theory for a large class of categories at once: monodial categories, topological categories, algebraic categories, locally presentable categories, etc. Of course, the same is true for other notions of category theory (functors, natural transformations, types of morphisms, etc.).

But if one has not heard of category theory before, the first motiviation should be to *think in categories* (in the colloquial sense). For example the set, which underlies a group, *really differs* from the group. In almost every book and lecture, this is absorbed by abuse of notation. The existence of bases in vector spaces is no reason at all to restrict linear algebra to vector spaces of the form $K^{(B)}$. Similarily, vector bundles should *not* be defined as bundles which are locally isomorphic to some $\mathbb{R}^n \times X$, which most topologists still ignore! Rather, it is first of all a vector space object in the category of bundles over $X$.

Let's conclude with an example which both introduces functors and **algebraic geometry**: Assume you have a system of polynomial equations $f_1(x)=...=f_n(x)=0$ in $m$ variables defined over $\mathbb{Z}$ and you want to study the solutions in arbitrary rings *at once*, using a *single mathematical object*, e.g. having in mind some local-global results of algebraic number theory. So for every ring $R$, we put $F(R) = \{x \in R^m : f_1(x)=...=f_n(x)=0\}$. Observe that for every ring homomorphism $R \to S$ there is a set map $F(R) \to F(S)$ and that this is compatible with composition of homomorphisms. This exactly means that $F$ is a functor from the category of rings to the category of sets. Algebraic geometry studies functors which locally look like the functor above.