the start state
The start state is where every computation begins, before a single symbol of the input has been read. It is the machine's 'I have seen nothing yet' state. In the five-tuple it is the element q0 of Q, and in the diagram it is the circle with a little arrow pointing into it from nowhere.
Because nothing has been read at the start, q0 must encode the truth about the empty input. That has a clean consequence: a DFA accepts the empty string ε (epsilon) exactly when its start state q0 is itself an accept state. If q0 is not in F, then a machine handed the empty input ends right where it began, in q0, and rejects. This is a small but common source of off-by-one mistakes when people design DFAs.
There is exactly one start state — never two, never zero. (This is one place DFAs differ from some NFA presentations, which can be more flexible.) When you design a machine, choosing q0 means deciding which fact about the history is true 'before anything happens', and getting that base case right is as important as getting the transitions right.
A DFA accepting strings with an even number of a-s starts in the Even state, because zero a-s (the empty string) is an even count. Since Even is also the accept state here, the machine correctly accepts ε. If you wrongly started in Odd, you would reject ε by mistake.
q0 encodes the empty input; a DFA accepts ε iff its start state is an accept state.
A DFA has exactly one start state. Do not confuse 'start' with 'accept' — a state can be both, one, or neither.