The Cinque Terre is a year-round destination, but the timing makes an enormous difference to the experience. July and August are by far the worst months to visit: trains are packed, the villages are overwhelmed with day-trippers, and the narrow lanes become unpleasant at midday. Prices are highest and accommodation books out months in advance.
April, May, and September through October are the best times. The weather is warm, the sea swimmable from May onwards, the trails are generally open, and the villages have something like their normal character in the mornings and evenings. Spring brings wildflowers on the terraces; autumn brings the grape harvest in the terraced vineyards.
November through March is quiet and often beautiful. Some restaurants and hotels close, the trails may be wet and slippery, and occasional storms hit the coast. But the light is excellent for photography and you will largely have the villages to yourself. The Christmas nativity scene in Manarola makes December worth considering.
The lemon season runs from May to October; the anchovy season peaks in late summer; the grape harvest in the terraced vineyards happens in September–October. All are good reasons to visit at those times.