Timbuktu is in a region in Mali named after the largest city within the region.
Whilst Timbuktu is the largest department in Mali, it is largely empty. The majority of Timbuktu is the deserts of the Sahara, the main towns are all focussed in the south along the Niger. The desert areas are largely the domain of the Tuareg.
The towns in the south can be accessed by bus/taxi. However the desert to the north requires a 4x4 or a camel, you will probably want a guide who knows what they are doing.
The city of Timbuktu itself is the main attraction in the department.
The world famous 'Festival in the Desert' is held at Essakane. You can easily get transport from Timbuktu when the festival is running.
If you are heading into the desert to the North there is usually considered to be a threat of bandits in addition to the risk of getting stranded. Be sure to register your itinerary and expected return date with the tourist office in Timbuktu.