A Boolean algebra $A$ is a complete Boolean algebra if for every subset $C$ of $A$ the arbitrary join and arbitrary meet of $C$ exist.

By de Morgan's laws, it is easy to see that a Boolean algebra is complete iff the arbitrary join of any subset exists iff the arbitrary meet of any subset exists. For a proof of this, see this link.

For an example of a complete Boolean algebra, let $S$ be any set. Then the powerset $P(S)$ with the usual set theoretic operations is a complete Boolean algebra.

In the category of complete Boolean algebras, a morphism between two objects is a Boolean algebra homomorphism that preserves arbitrary joins (equivalently, arbitrary meets), and is called a complete Boolean algebra homomorphism.

Remark Between a Boolean algebra and a complete Boolean algebra, there are many intermediate concepts. Let $\kappa$ be a cardinal. A Boolean algebra $A$ is said to be $\kappa$ complete if for every subset $C$ of $A$ with $|C|\le \kappa$ $\bigvee C$ (and equivalently $\bigwedge C$ exists. A $\kappa$ complete Boolean algebra is usually called a $\kappa$ algebra. If $\kappa=\aleph_0$ the first aleph number, then it is called a countably complete Boolean algebra.

Any complete Boolean algebra is $\kappa$ complete, and any $\kappa$ complete is $\lambda$ complete for any $\lambda\le \kappa$ An example of a $\kappa$ complete algebra that is not complete, take a set $S$ with $\kappa < |S|$ then the collection $A\subseteq P(S)$ consisting of any subset $T$ such that either $|T|\le \kappa$ or $|S-T|\le \kappa$ is $\kappa$ complete but not complete.

A Boolean algebra homomorphism $f$ between two $\kappa$ algebras $A,B$ is said to be $\kappa$ complete if $$f(\bigvee \lbrace a \mid a\in C\rbrace)= \bigvee \lbrace f(a)\mid a\in C\rbrace $$ for any $C\subseteq A$ with $|C|\le \kappa$