I should start of by noting that we do not know a lot about the Huns so everything I state can be, and possibly will be argued against.

The "black" Huns (although I must say I have never heard them being referred to as such) were a nomadic group of barbarians from the Russian steppe, specifically we first hear about them being located (possibly) east of the Volga river from Tacitus.

They are most famous for pillaging their way across the Roman Empire in the 5th century while being lead by Attila although they were eventually defeated at the hands of the Germans and Romans at Chalons and Nedao and after that receded into the shadows of history as quickly as they came.

Now for what I actually know about the black Huns I am gonna use "John Man: Attila the Hun" as a source

The black Huns were probably Turkic in origins, its quite difficult to tell because they picked up a lot of stragglers on their rampage westwards, so who is and who isn't a Hun can and is really fuzzy. However, if we look at the name of some of Attila's relatives we might get a clue, from John Man: Attila The Hun page 73.

"But we have Hun names, or think we do [..] Attilla's uncle Octar was also written Oiptagos, Accila, Occila, Optila and Uptar (ct shifted to pt in the Balkan dialect of Latin). but Öktör means "powerful" in old Turkish. A coincidence? Scholars think not. The names of other characters in this [Attila's] story also suggest Turkish roots: Attila's father Mundzuk means Pearl or Decoration, his Uncle Aybars means Moon Panter and his senior wife Erekan means Beautiful Queen, his son Ernak means hero, a shadowy king Charaton/Kharaton means Black, and the -kam ending on a few Hunnic names seems to recall the Turkish for Priest or Shaman"

Now the phenomenon of the white Huns occur a few decades later, they [The White Huns] hang around the area of Bactria, Transoxania and Afghanistan, if you look on map today and notice an area pack-filled with places named -stan to the north east of Iran, you found the place.

They start to make their mark on history in the 430's when they become a thorn in the side of the Sassanian Persians and actually manage to make them pay tribute in 483-485 AD. They are however, ultimately driven out of Persia and routed at the hands of Khorsau I in 557, after which they fell under the wing of the Göktürks.

Now whether the white Huns are related to the black Huns is uncertain and argued about.

According to Procopius's History of the Wars, written in the mid 6th century - the Hephthalites "are of the stock of the Huns in fact as well as in name: however they do not mingle with any of the Huns known to us. They are the only ones among the Huns who have white bodies...."

However, the modern consensus seems to be leaning on no, from Enoki, Kazuo: "On the Nationality of the White Huns"

There are numerous debates about Hephthalite [Another name for White Huns] language. Most scholars believe it is Iranian for the Pei Shih states that the language of the Hephthalites differs from those of the Juan-juan (Mongoloid) and of the "various Hu" (Turkic); however there are some think the Hephthalites spoke Mongol tongues like the Hsien-pi (3rd century) and the Juan-juan (5th century) and the Avars (6th-9th century). According to the Buddhist pilgrims Sung Yun and Hui Sheng, who visited them in 520, they had no script, and the Liang shu specifically states that they have no letters but use tally sticks. At the same time there is numismatic and epigraphic evidence to show that a debased form of the Greek alphabet was used by the Hephthalites. Since the Kushan was conquered by Hephthalites, it is possible they retained many aspects of Kushan culture, including the adoption of the Greek alphabet.

Sources: Enoki, Kazuo: "On the Nationality of the White Huns" and John Man: Attila The Hun