If you take into account the way a Boolean circuit can change with time, then you have to come to the conclusion that you can build any computational machine using just one type of logic gate. Using nothing but a universal gate you can build combinatorial logic and active devices such as flip-flops and hence the components of memory.
You can see that you can build a finite state machine from nothing but universal logic gates. You can even build a Turing machine, as long as you simulate its tape using memory built from universal logic gates. Of course, in practice, the machine’s memory would be finite and bounded and so you have really just another finite state machine.
It is remarkable that something capable of so much complex behavior as a computer can be built from the most basic of components - the universal logic gate.
Summary
George Boole invented a system of logic that encapsulated the way true and false changed when combined using AND, OR and NOT.
Boolean logic can be represented by listing the results of all inputs in a table or as an equivalent formula.
Boolean operators and more generally Boolean functions can be implemented as electronic modules usually called "gates".
You can think of the two values, true and false, as any two states. In particular you can think of them as one and zero, i.e. binary.
Boolean operators can be used to implement binary arithmetic and other binary functions.
Real logic gates need time to change state and this fact can be made use of to implement devices that aren't simply static logic functions.
A computer is built using logic gates that change state, usually under the control of a clock pulse.
Logic is used in programming but it isn't as easy as at it seems. In particular De Morgan's Laws summarize the relationship between expressions involving AND and OR.
A universal gate or operator is one that can be used to implement any Boolean function. There are a number of universal gates and they all involve negation.
It is possible to build a complete computer using nothing but multiple universal gates.
