For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.
The idea of Lebesgue integral is the following: we give to a simple non-negative function $\sum_{j=1}^Na_j\chi_{S_j}$, where $a_j\geq 0$ and $S_j>0$ the value $\sum_{j=1}^Na_j\mu(S_j)$. Then we define the integral of a measurable non-negative function as $$\int_X f(x)d\mu(x):=\sup\left\lbrace \int_X g(x)\mathrm{d}\mu(x) \mid 0\leq g\leq f,\ g \text{ simple}\right\rbrace.$$ For a measurable function, write $f=\max(f,0)-\max(-f,0)$ to give a value to $\int_X f(x)\mathrm{d}\mu(x)$.
The major interest is that we can integrate functions which are defined in an arbitrary set, provided we have fixed a $\sigma$-algebra and a measure on it.
When dealing with a function $f\colon[a,b]\longrightarrow\mathbb R$, with $a,b\in\mathbb R$ and $a\lt b$, the Lebesgue integral is more general than the Riemann integral: if a function is Riemann-integrable, then it is Lebesgue-integrable (and the integrals are the same), but there are functions (such as characteristic function $\chi_{[a,b]\cap\mathbb Q}$) which are Lebesgue-integrable, but not Riemann-integrable.
