Tanzania's climate varies across the country, from the tropical humidity of the coast, the dry heat of the savannah and the central areas of Katavi and Ruaha and the cool air of the Ngorongoro highlands and Mt. Kilimanjaro.
As with the rest of East Africa most parts of the country experience two rainy seasons: the 'long rains' falling over a ten week period between April and June, and the 'short rains' over a five week period between November and December. Most camps close during the long rains for refurbishment and a lot of the roads become impassable and so it is best to avoid travelling during this time.
As a general rule the best time to visit the southern parks of the Selous and Ruaha is between June and October the water dries up and game viewing is at its peak (although January through to the end of March is also good as the landscape is beautifully green and it is slightly quieter).
If visiting the northern circuit then January to March sees the wildebeest migration arrive onto the southern plains of the Serengeti to calve before heading north west between the months of June and October.
In the west Katavi is at its best between August and October when water is very hard to come by and the animals congregate around the limited numbers of rivers and streams thus making for exceptional game viewing. Mahale is good all year round but to be there in cooler conditions (more comfortable if you have to trek a long distance to see the chimps) then the months of August to October and then January to March are preferable.
Finally the coast typically follows the general pattern of the rainy seasons and so the windows of June to October and mid-December to March are the best, the latter being a little more humid as the heat intensifies before the long rains.