How to explain Borel sets and Stieltjes integral to beginner maths student? The problem is that I know by the definition what Borel sets and Stieltjes integral are but I'm not good to explain in layman terms what they are. Is there easier answer that "write down the definitions until you have reduced everything to the axioms"?

A Borel set is a set that we can create by taking open or closed sets, and repeatedly taking countable unions and intersections. Any measure defined on a Borel set is called a Borel measure. 
The Riemann integral is a definite integral that was the foundation of integration over a given interval (before the far superior Lebesgue integration took over). The idea of Riemann integration is to approximate the region you are given, and continuously improve the approximations. We take a partition of an interval 
\begin{equation*}
a=x_0<x_1<...<x_n=b
\end{equation*}
on the real line $\mathbb{R}$, and approximate the area we are integrating with "strips". As the partition gets finer, we take the limits of the Riemann sum (the area of each of the strips) to get the Riemann integral. The Stieltjes integral is a generalisation of this. It can be represented in the form
\begin{equation*}
\int fd\mu 
\end{equation*}
with respect to the measure $\mu$ (however there must be no point of discontinuity). 
Does that help? 
