The simplest and earliest example I know regarding the renormalization group idea is the following.
Suppose we want to study some feature $\mathcal{Z}(\vec{V})$ of some object $\vec{V}$ which is in a set $\mathcal{E}$ of similar objects. Suppose that unfortunately this question is too hard. What can one do?
The renormalization group philosophy is try to find a "simplifying" transformation $RG:\mathcal{E}\rightarrow\mathcal{E}$, such that $\mathcal{Z}(RG(\vec{V}))= \mathcal{Z}(\vec{V})$, and $\lim_{n\rightarrow \infty} RG^n(\vec{V})=\vec{V}_{\ast}$ with $\mathcal{Z}(\vec{V}_{\ast})$ easy to understand.
Example (Landen-Gauss, late 1700's):
Let $\vec{V}=(a,b)\in\mathcal{E}=(0,\infty)^2$ and consider $$ \mathcal{Z}(\vec{V})=\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{a^2 \cos^2\theta+b^2\sin^2\theta}}\ . $$
A good choice of renormalization transformation here is $RG(a,b)=\left(\frac{a+b}{2},\sqrt{ab}\right)$, as discovered by Gauss.
A recent example now.
Example (Kadanof-Wilson, late 1960's early 1970's):
Take $\mathcal{E}$ to be the set of Borel probability measures on $\mathbb{R}^{\mathbb{Z}^d}$. Let $\mathcal{Z}(\vec{V})$ be equal to $1$ if the two-point function decays exponentially and $0$ otherwise.
Then define $RG$ as the transformation which gives the law of the block-spinned/coarse-grained field as a function of the law of the original field.
Note that the feature $\mathcal{Z}$ that one would like to preserve can be defined a bit more loosely. One could, e.g., "define" it as the long-distance behavior of the random spin field with probability law $\vec{V}$ (or in physics jargon: the low energy effective theory). Also in the dynamical systems context, it could be the chaotic behavior or not of a map $\vec{V}$. Then $RG$ could be a doubling transformation, i.e., composition of the map with itself together with some rescalings and reversals of orientation.