as the title suggests, I was struggling to define the differences between finite and Bchi automata and how they are represented. From an assignment I'm working on, this automaton can be depicted as both infinite automaton and Bchi automaton.Can you give me an example for each of those automatons to understand the differences? 
What is the difference between finite automata and Bchi automata?
automata;finite automata;buchi automata
The automaton models themselves, that is the syntax, are indeed identical: both have a finite set of states, a transition relation, and initial and final state(s).The difference lies in the acceptance criteria, that is the semantics.A finite automaton $A$ accepts a word $w$ if and only if there is a computation for $w$ in $A$ that ends in a final state.A Bchi automaton $B$ accepts a word $w$ if and only if there is a computation for $w$ in $B$ that visits final states infinitely often.The differences follow from this.For which there is a formal definition!Other acceptance criteria have been proposed, e.g. visit one final state infinitely often.