To gain entry into Tanzania, US citizens and most other nationalities will need a passport and visa. The passport must be valid for 6 months after the intended length of stay.
Visas can be obtained prior to departure from the USA (recommended due to long wait times at JRO) or at your point of entry into Tanzania. The visa cost for US citizens is $100. More information can be found here: http://tanzaniaembassy-us.org/?page_id=76#visaForms
Obtaining a Visa at JRO airport for citizens of most countries is a relatively simple process. When you arrive at JRO, as you enter the airport there will be two lines. The line on the right is for people purchasing a visa. The line straight ahead is for people who have a visa. To get your visa you fill out one form, show them your passport and pay $100 USD (If you are a US citizen).