DFA (Finite Automata), please help me to understand

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Indicate the state that the DFA will end in after processing each of the following input strings. Note: the input labeled "The empty string" is literally the empty string—a string with no letters in it—not the letters 'T', 'h', 'e', ' ', 'e', and so forth.

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For the string = abcba,

do I end in state 2? Also, what is that double circle means?

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From https://www.tutorialspoint.com/automata_theory/deterministic_finite_automaton.htm

The vertices represent the states. The arcs labeled with an input alphabet show the transitions. The initial state is denoted by an empty single incoming arc. The final state is indicated by double circles.

And I'm not really sure but I think you stop at 2.

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Yes, after abcba, you end in state 2.

The double-circle usually indicates an accepting state -- in the DFA for a regular expression, the string you've received so far matches the regular expression iff you're in an accepting state.

If a regular expression matches the empty string then the start state will also be an accepting state, as is the case here.

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