Citizens of most countries require a valid passport, and a Tanzanian visa to enter the country.
The visa seems to be more of an entry tax than a way to screen out undesireable types... you pretty much just hand over the cash and get your passport stamped. You can obtain a visa in advance at the Tanzanian embassy in your home country (this is what the Tanzanian government advises), but it's much simpler (and cheaper!) to get the visa on arrival in Tanzania. At the airport in Arusha, there is a separate line for visitors without visas. Hand over $50 US (cash only!) and your passport gets stamped. When we went, the line for visitors without visas was actually shorter than the line for those who had obtained them in advance!
Warning: it's worth checking this info with your local Tanzanian embassy, or the
Government of Tanzania website, as policies and procedures may change from time to time.
Tanzania now requires visitors arriving from a country that has a risk of yellow fever to show proof of immunization at the point of entry. This includes people arriving from neighboring countries such as Kenya. Even if you're flying direct into Tanzania, the yellow fever vaccination is recommended. Check out the Health and Fitness page for more details on immunizations.
We've heard that the yellow fever immunization is available (for a fee) on arrival at the Kilimanjaro and Dar es Salaam airports, but it's better to get the vaccine before you leave home.