The Thai groups have all been classed as speaking a form of the Tai-Kadai group of languages.  The Thai language is tonal, like many Sino-Tibetan languages.  It was thought at one time that the different Tai langages were also a branch of the Sino-Tibetan family tree, but this is no longer thought to be true. 

The Black Thai speak a language called Tai Dam.  The Red Thai speak Tai Daeng, but seem (confusingly!) to prefer the name "Tai Dam" for themselves.  The White Thai speak a related languge -- Tai Don.  

This means that the Thai languages are linguistically unrelated to Vietnamese, the national language of the country.