a state
A state is one of the situations the machine can be in — one circle in the diagram, one entry in Q. The whole point of a finite automaton is that a state captures everything the machine needs to remember about the input it has read so far. If you tell the machine which state it is in, it does not need to look back at the input it already consumed; the state is a perfect summary of the relevant past.
Here is the way to think about it when you design or read a machine: each state should stand for one distinct fact, or one combination of facts, about the history so far. In the 'ends in 1' machine, state A means 'the last symbol I read was not a 1 (or I have read nothing yet)' and state B means 'the last symbol I read was a 1'. That is literally all the machine remembers. Two different input histories that lead to the same state are, as far as the rest of the computation is concerned, indistinguishable.
Because the set of states Q is finite, the machine can only ever distinguish finitely many situations. This is both its elegance — you can list every situation — and its limit: any property of the input that would require telling apart infinitely many situations (such as 'the number of a-s read so far', which can be 0, 1, 2, 3, ... without bound) cannot be tracked by a finite set of states.
To accept binary strings whose value is divisible by 3, use three states r0, r1, r2 meaning 'value-so-far leaves remainder 0, 1, or 2 when divided by 3'. The remainder is all you must remember; r0 is the accept state. Three facts, three states.
A state = one fact (or combination of facts) about the input read so far.
A state is not a value stored in memory you can do arithmetic on; the machine cannot add to a state or compare two states. It can only be in one state and move to another according to δ.