The primary language of Tibet is actual Tibetan, but since it is derived from the many dialects of Chinese, you are more likely to get by if you can understand some Mandarin. This will work for most everybody you meet except some of the nomadic tribes found in the Far East regions of the country. The language is somewhat related to Burmese and only distantly related to Chinese, but depending on the dialect spoken, it can take on either a tonal or non-tonal form, and you need to take that into consideration.
In the big cities most of the people are fluent in Chinese, and that should tell you a lot about the culture and the politics of the region. Most of the signs including the traffic signs, are bilingual – Chinese as well as Tibetan. Often there will be a third if a major local language is spoken there. Clearly, Chinese is the most useful language for travelers to Tibet, and that should be your main clue as to what you want to brush up in anticipation of your next trip to this mysterious and wonderful land.