Annapurna is the 10th highest mountain in the world and was the first 8000m peak to be climbed. It was first summited in 1950 by Maurice Herzog and Louis Lachenal. It was not until 1970 that Annapurna was summited again. In the autumn of 1999 there were 120 expedition to Annapurna . 106 climbers reached the summit. 54 climbers never came back.
However, I am assuming you are more interested in trekking the Annapurna region , so most of this information will pertain to trekkers considering this trip.

The best season to climbing is usual trekking season in Nepal is from October to May. The rest of the year is the monson season and is not desirable for either trekking or expeditions. Besides the cold and rain, leeches are a problem during the monsoons. If you are going to be trekking to Annapurna , October and November are the best months.

Trekking possibilities are quite varied, but most will start at Pokhara. The circuit trek to the Annapurna Sanctuary will take at least 10 days, but allow 14 to have time to enjoy it and to allow for unforseen events. The route up the narrow Modi Khola river valley is quite wet and slippery and quite steep in a few places. It can be a bit dangerous. The trail up to the Annapurna basecamp is a more serious undertaking.