Tamashek is the name given to the macrolanguage comprised of the four closely related languages spoken by the various Tuareg peoples. More particularly, it is the name for the macrolanguage specified by the ISO-639-2 and ISO-639-3 standard code tmh.
You could say without reservation that Tamashek is the language of the Tuareg people. However, you could also very accurately say that no Tuareg actually speaks a language called Tamashek! Instead they speak one or the other of the languages: Tawallammat, Tayart, Tamasheq, or Tahaggart.
However in each of these languages there is a word to speak of the larger language group of which their language is a part. When referring to the larger language group, Tawallammat speakers use the word Tamajaq, Tayart speakers use the word Tamajeq, Tamasheq speakers use the word Tamasheq, and Tahaggart speakers use the word Tamahaq.
So Tamashek is a word invented by linguists to specify the macrolanguage which all Tuaregs claim to speak, even though they ironically use differring words to speak of it.