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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finite automaton is the graphical representation of language and regular grammar is the representation of language in expressions
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).
This is called a discrete set (all points isolated) or a finite set. Finite sets are always discrete.
It is a finite number.It is a finite number.It is a finite number.It is a finite number.
There are a finite number of apartments. Finite numbers may be large or small. There are a finite number of states. The number of molds in my fridge is not exactly finite.