The best time to trek to Annapurna Circuit is just after the monsoon is over in September, Nepal’s monsoon season last till end of August; sometime it last until first week of September, monsoon rain clears up all the dust and rainy season brings greenery in low foothill, and view of white peaks are super when we have blue sky. No snow on higher places and Throng La Pass is easily crossable. September, October and November is peak time of Annapurna Circuit trek.
Second best time to trek to Annapurna Circuit is spring (Mar to May), though there are chances of soft rain and thunder storm, sky can be hazy sometimes in lower place however there are haze in the mountains. The benefit of trekking in April and May are warm temperature or even begging of June is okay to trek to Annapurna.
Trekking in winter (December to February) can be risk of snowfall, if snowfall is heavy then pass is blocked and you may have to turn back same way down to Besisahar. It is cold in winter, the night and early morning temperature can be around negative 14 degree Celsius outside the room. In Nepal we have no certain time if snowfall in Himalaya; thus it can be anytime in dry season (October to March)
Still you need to bear in mind that climate change and global warming are affecting our world and the weather condition in Himalaya changes fast so there are chances are rain and weather turn bad even in peak seasons.
And Annapurna Circuit trek is best alternative for those having vacation in summer (June to August). Upper part of Annapurna Circuit, mostly Manang Valley and Mustang is said be under the rain shadow zone. However; monsoon rain may affect the tails and roads / bridges are washed away sometimes, so it is always wise to allow few reserve days just in case bad weather, landslides and respectively flight delays that changes your plan.