Basically, the Tibetan climate is not as harsh as many people imagine it to be. The best time of year to be in Tibet is from April to the beginning of November, after which temperatures start to plummet.
The central Tibet, including Lhasa, Gyantse, Shigatse and Tsedang, generally has very mild weather from April to November, though July and August can be rainy - these two months usually see around half of Tibet's annual rainfall.
October and November often bring some dazzling clear weather and daytime temperatures can be quite comfortable at Tibet's lower altitude.

The coldest months are from December to February. It is not impossible to visit Tibet in winter. The low altitude valleys of Tibet (around Lhasa, Shigatse and Tsedang) see very little snow.
Spring does not really get under way until April, though March can have warm sunny days and is not necessarily a bad month to be in Tibet. More specific information in different areas:

Lhasa - the border of Nepal/China: The Friendship highway is basically in good conditions year around. But from December to February, the thawed road could make some trouble Besides, try to avoid August - landslide could happen in the rainy season.

Mt. Everest Region: Early May and early October are the best time to visit Mt. Everest. Due to the clear weather, you have great chance to see Mt. Everest's true face. From December to February, it's too cold to go to this region. But the magnetism of Mt. Everest always attracts people anytime of the year.
Ali (Mt.Kailash): Even without climate restrictions, this area is already inhospitable. Big rain and snow could make the journey worse. However, for those determined tourists, the appropriate time is May, June, July, September and October.
Eastern Tibet: Don't touch this area in July or August, the rain could ruin the road, and make terrible landslides. While in winter, the road could be frozen.

Northern Tibet: With the average altitude of 4,500m, this area offers very limited time for tourists. Summer is the prime time to enjoy the great plain in northern Tibet.