Here the same word is a countable noun naming a person. You can say a Tibetan, two Tibetans, many Tibetans. Because it is countable, it needs an article in the singular: he is a Tibetan, not he is Tibetan-noun with nothing in front. In everyday speech people often use the adjective instead and say 'she is Tibetan', which means the same thing and sounds lighter. Keep the capital T in both. The plural is regular: Tibetans.

This is the language sense, and it is uncountable. That changes the grammar in two ways. You do not put a in front of it, and you do not add s: say speak Tibetan, learn Tibetan, in Tibetan. If you want to make it sound more formal or more precise, you can say the Tibetan language, where Tibetan goes back to being an adjective. English names most languages with this same shape, so the word for the people and the word for their language often look identical, and only the grammar around them tells you which one you are reading.