It seems to me that this topic is a main feature of modern mathematics, if not its characteristic feature, compared with classic mathematics: moving the complexity from individuals to the species. A complicated property in a simple situation becomes a simpler property in a suitable more complex setting, and the trade-off can be very convenient.

The easiest and oldest example I have in mind is the Cartesian product: instead of talking of two or more objects $a\in A$, $b\in B$ and $c\in C$ we can consider just one object $(a,b,c)$ in the possibly more complicate setting $A\times B\times C$ (so that, for instance, we can describe a system of equations in $\mathbb{R}$ as a single equation in $\mathbb{R}^n$ ). This approach is so powerful because modern mathematics exploits the axiomatic method to its further consequences: working in a more complicated setting does not necessarily entail more difficult methods. What is needed is an abstract point of view, to identify the exact hypotheses that make a method work, and to understand to what extent these hypotheses pass to the more complicated constructions.

To remain in the example of the equations above, for instance, the Contraction Principle passes from $\mathbb{R}$ to $\mathbb{R}^n$ or even more complicated structures with no formal difference in the statement and in the proof; the key point is that $\mathbb{R}^n$ is complete just as it is $\mathbb{R}$. To make one more example in the same spirit, a linear system of ODE with constant coefficients is written $\dot u = A u$ and solved as $u(t)=e^{tA} u _ 0$, exactly as the more elementary scalar equation $\dot u = a u$, and from the same archeotypal equation the whole theory of evolution equations spring off, by which it is possible to treat evolution equations and PDE's.

The acme of this method is reached in Category Theory, that expressly created a language to construct more and more complex settings, where every universal concept can find a simple formulation. Here's one last example: a rather complicated situation as it is an adjunction of functors between two categories, can be seen as an "initial object", the most simple categorical notion, in a more complicated category, which is the starting point to Freyd's existence theorem for adjoint functors (*edit:* for details on this example check Todd Trimble's answer to a further question).