A boolean algebra B is a complete lattice with the following properties:
For any a,b of B let a and b := inf{a,b} and a or b:=sup{a,b}.
1. Distributivity: a and (b or c) = (a and b) or (a and c)
2. Distributivity: a or (b and c) = (a or b) and (a or c)
3. Existence of a negation operation not: B -> B with
1. ((not a) and a) or b = b
2. ((not a) or a) and b = b

From these follows:

• all other calculation rules (some follow directly from the lattice)
• the unique existence of a 1 and 0 (in fact 1=sup(B) and 0=inf(B)), one could also define 1 and 0 and their properties in the definition
• boolean algebras doesn't just consist of 0 and 1, they can be rather strange in the infinite case
• there is some connection to the algebraic definition of an algebra
• in a finite boolean algebra every element can be constructed from minimal elements, the so called atoms. This is the basis for the representation of boolean functions by disjunctive normal forms

Examples: set of subsets of a set.
the set of functions from an arbitrary set into a boolean algebra