Culturally Tibetan areas are in five modern provinces of China - the Tibet Autonomous Region (TAR), Sichuan, Qinghai, Gansu and Yunnan. One needs simply a standard Chinese visa, which can be procured from a Chinese embassy or consulate. However, if you plan to travel to the TAR, which includes Lhasa, you ALSO need to procure a special travel permit. Such special permits are not required for the other four provinces. Every year there are rumours that the permit system will be eliminated, but as of the summer 2007 it persists.
Of course if you are going to the TAR in an organized tour, the tourist agency will deal with all of these permit issues. Otherwise, travel agents in Chengdu can usually quickly arrange a "tour" for you in the matter of a few days and you can then stay within Lhasa as long as your visa is valid. Such travel agents can be found in and around the Traffic Hotel in Chengdu. It usually costs up to $100 beyond the cost of the hotel and air flight for them to arrange such a permit. However, keep in mind that such a permit just allows you to go to Lhasa and anywhere else the tour itinerary specifies - it does not give you permission to travel anywhere at all in the TAR. Individual tourists do take their chances and travel places without permits, but you won't find a driver willing to drive you and you may run into problems.
Unfortunately, if you try to get a visa and travel permit from Nepal, you will find that you are only given a visa for the duration of the tour. It is thus easy to get a two week or so permit and visa, but if you want to stay in Tibet or China longer than that you will be out of luck.