the unique minimal DFA
Among all the DFAs that recognize a given regular language, there is a single smallest one, and it is essentially the only one of that size — a canonical 'best' machine for the language. Think of it like the reduced form of a fraction: 2/4, 3/6 and 50/100 all denote the same value, but 1/2 is the unique simplest representative. Likewise, many DFAs can accept the same language, but exactly one (up to merely renaming the states) has the fewest states.
Precisely, every regular language L has a minimal DFA that is UNIQUE up to isomorphism — that is, unique except for what you call the states; the structure, transitions, and accept set are forced. Its states correspond one-to-one with the Myhill-Nerode equivalence classes of L, so its state count equals the index of L. You obtain it either by running minimization (remove unreachable states, then merge all equivalent ones) or directly by building one state per indistinguishability class. Two reachable, distinct states in the minimal DFA are always distinguishable, by construction.
Uniqueness is what makes the minimal DFA the CANONICAL form of a regular language, and that has real payoff. Equivalence of two regular languages becomes decidable and routine: minimize both DFAs and check whether the results are the same machine (isomorphic). It also gives the exact answer to 'what is the smallest automaton possible?' — no heuristic, no guessing. Honest caveat: this uniqueness is special to DETERMINISTIC finite automata; minimal NFAs are NOT unique and finding a smallest NFA is computationally hard.
Two engineers draw different 5-state DFAs for 'binary strings divisible by 3'. Minimizing both yields the SAME 3-state machine (states for remainder 0, 1, 2) up to renaming — the unique minimal DFA. So the two original machines accept the same language.
Different DFAs for one language minimize to the same canonical machine.
'Unique' means up to ISOMORPHISM — renaming states does not count as a different machine. This canonical-form property is special to DFAs: minimal NFAs need not be unique, and minimizing an NFA is PSPACE-hard, not a simple table-filling pass.