Borel $\sigma$-Algebra definition. 
Definition:
The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals $\ (a, b]$, where $\ a<b$ in $\mathbb R$ (We also allow the possibility that $\ a=-\infty\ or \ b=\infty$) Its elements are called Borel sets.
For A $\in$ B($\mathbb R$), the $\sigma$-algebra
$$B(A)= \{B \subseteq A: B \in B(\mathbb R)\}$$
of Borel subsets of A is termed the Borel $\sigma$-algebra on A.

I struggle with this part especially "generated by the $\pi$-system $\mathcal J$ of intervals (a, b]"
In addition could someone please provide an example of a Borel set, preferably some numerical interval :)
Also is $\mathbb R$ the type of numbers that the $\sigma$-algebra  is acting on?

Borel $\sigma$-algebras turn op if we are working on topological spaces. If $X$ denotes a topological space and $\tau$ is its topology then the smallest $\sigma$-algebra that contains $\tau$ (in other words: the $\sigma$-algebra generated by $\tau$) is the Borel $\sigma$-algebra on that space. In special case $X=\mathbb R$ equipped with its usual topology it can be shown that the $\sigma$-algebra generated by the open sets (by definition the Borel $\sigma$-algebra) coincides with the $\sigma$-algebra generated by intervals $(a,b]$. 
