A Buchi automaton is a regular automaton but reads infinite words instead of finite words. A word is defined to be in the language of the automaton iff a run of the automaton on it visits inifinitly many times in the group of final states (or receiving states).
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Infinite word automatons, invented by Buchi in the sixties, were used for reasoning about logic. Today they are used for formal verification of systems.
finite automaton is the graphical representation of language and regular grammar is the representation of language in expressions
I would guess that is because it has a finite number of different states. (It is also known as a finite-state machine.)
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