Zanzibar is a semi-autonomous archipelago off the coast of Tanzania consisting of Zanzibar Island (locally, Unguja), Pemba Island and Mafia island.
There is a lot to see and to do on Zanzibar island. First and foremost, do enjoy the amazing white sand beaches especially at the north, east and south coast. The sand has the consistency of castor sugar! Zanzibar also has quite a lot of caves, e.g. Tazani near Nungwi. A tour company in Jambiani "Mambopoatours" offers excursions off the beaten track. Also do a village tour that are offered all over the island. The villagers don't see tourists as intruders, to the contrary, they welcome foreigners. Please do make sure that you are dressed with a t-shirt and knee-length pants - Zanzibarians are 98 % Muslims and walking through villages in beach gear is disrespectful.
Jozani Forest has excellent nature trails, featuring some very exotic (and large) trees and plants. Even more interesting, though, are the Red Colobus Monkeys that live here. These Monkeys can only survive on Zanzibar, nowhere else in the world, since they need a diet of 70 different plants, berries etc. The Red Colobus Monkeys are a protected species. A major part of the entrance fees goes to the local farmers in the surrounding area. In the past, the farmers killed the Monkeys because they destroyed their crops. Ever since they are compensated for their losses, the killing stopped. They are very curious and playful and will likely pose for a picture. The entry fee (USD10) also include an optional visit to a beautiful mangrove forest which is highly recommended. You can take the local daladala from Zanzibar Town for TZS 2000 per person. They might want to charge you more (because you are a tourist).