Shoulder season is the month or two either side of a destination's peak — typically April–May and September–October in Europe. You get most of the good weather at a significant discount, with thinner crowds and everything still open.
Shorter days, some chance of poor weather, and — in seasonal destinations — reduced ferry schedules, closed mountain roads or shut hotels. September holds summer sea temperatures around the Mediterranean; May does not, so a beach trip and a city trip have different shoulder windows.
Where the whole point is seasonal: island-hopping outside the ferry schedule, hiking before the snow clears, or northern lights in a month with no darkness. Check what actually runs before booking the cheap month.