Positive for both calls and puts - more volatility means more chance of a large favourable move, with the unfavourable side capped.
Largest at the money and for longer-dated options. That gives the practical rule: to trade a view on volatility itself, use long-dated options where vega dominates. To trade a view on a specific event, use short-dated options where gamma dominates.
Not a Greek letter, which occasionally comes up as trivia - it is the only member of the set that is not.