In The Beginning... There was 1 and 0
Boolean Algebra is a two state system of algebra, we can apply this to the binary number system where only two valid numeric symbols exist, "1" and "0" and thus use boolean logic to solve binary problems. The electronics inside a computer rely on this system, modeling voltage in a connection as either on (1, or TRUE) or off (0, or FALSE).
Boolean Operators
In Boolean Algebra there are three fundamental operators (although more do exist, however any possible equation can be represented by a combination of these fundamental operators).
AND
A•B = C is the same as saying "A and B equals C", the value of C can be calculated based on a table of possible values of A and B, known as a Truth Table.
AND is the boolean equivalent of multiplication, C will be true if, and only if both A and B are also true.
OR
A + B = C is the same as saying "A or B equals C", as with all boolean algebra the answer to this equation can be shown by the use of a truth table.
OR is the boolean equivalent of addition, C will be true if A, or B, or both are true.
NOT
is the same as saying "not A equals B", this operator is only has one variable, and so its truth table is much simpler.
NOT is the boolean equivalent of inversing or complementing, if A is true, B will be false, and if A is false, B will be true.
Logic Gates
We can think of these operators as being part of a circuit, where a signal is carried along lines - or wires in the circuit - through the operators, which are known on a circuit diagram as logic gates. The symbols to represent the different logic gates are shown in the example diagram below:
References:
Janet Lavery - Durham University Computer Systems, Machine Architecture Lecture 1, 2007