Given a set $\Omega$, a $\sigma$-algebra on $\Omega$, $\mathcal{A}$ is a subset of the Power set of $\Omega$ ($\mathcal{A} \subset 2^\Omega$), s.t.

1. $A$ is non-empty.
2. $A$ is closed under complements. $E \in \mathcal{A} \Rightarrow E^c \in \mathcal{A}$
3. $A$ is closed under countable unions. $E_1, E_2, ... \in \mathcal{A} \Rightarrow \bigcup\limits_{i=1}^{\infty} E_i \in \mathcal{A}$

(PP 1.2) Measure theory: Sigma-algebras

From these axioms, one can show that a sigma-algebra is closed under countable intersections too.

The sigma-algebra generated by $C \in 2^\Omega$, written as $\sigma(C)$, is the "smallest" sigma-algebra containing $C$. See here to see precise definition and why this always exits.

A common example is the Borel sigma-algebra.

A sigma-algebra can be generated by an algebra, as explained in the Caratheodory extension theorem