When to visit

You can visit Tanzania year-round, also depending on the places you like to visit. 

The best time to visit Tanzania is between June and October, when there are no rains and the temperatures are cool. But at this time the hotels and lodges will often be full and airfares are high. 
Another good time to visit the country is from late December to early March, just between the short rains and the long rains, but temperatures will be higher.
 If you want to avoid a lot of other tourists, and/or if you are on a tight budget, you should conisder travelling in the rainy season from March untill June. Because of the rains, this is also the 'green' season, which can be very attractive. But be aware that in the rainy season it can be difficult to travel because of muddy roads, and some of the lodges and hotels are closed during this time.

It is better to book as far in advance as possible to ensure availability of rooms and other service.

This are the best times for some of the places:

Place When to visit
Arusha NP Almost year-round
Kilimanjaro Year-round except April - May (rains)
Serengeti November - July (migratory animals are in Serengeti usually from November untill May/June
Ndutu (Serengeti plain) December - February
Ngorongoro Almost year-round
Lake Manyara NP December - June
Tarangire NP July - November
Selous / Ruaha July - end of February
Zanzibar Year-round except March - May (rains)