It's a mixed bag really. In most cases, the current names used in English come from other languages a third-party language or sometimes through even longer games of linguistic telephone.

For example, English Sioux is an abbrevation of the French Nadouessioux, which in turn came from either Ojibwe or Odawa from a root referring generically to non-Algonquian-speakers and/or snakes (something akin to natowewa, from which the name for the Iroquian-speaking Nottoway people is also derived). The people the Ojibwe referred in this case, and the ones English refers to with the term "Sioux", referred to themselves as the Oceti Sakowin, the Seven Council Fires. Generally today, the members of the Oceti Sakowin are referred to as the Dakota and the Lakota. You can see an outline of the internal divisions here; some of the names of which were also imported into English.

For the Cherokee, the situation is a bit more ambiguous. There's some debate over whether English got the word from Tsalagi, ciloke celokke (fixing spelling error), or some other word. Tsalagi is the Cherokee word for their own people, as seen in the name of the Cherokee Nation (Tsalagihi Ayeli) and the Cherokee language (Tsalagi Gawonihisdi). The Mvskoke (Muscogee) word ciloke celokke (pronounced with a chi sound at the beginning) refers to non-Muscogean-speakers, and generally referred to either Iroquoian or Souian-speakers in the Carolinas. From this word (or a close linguistic relative) de Soto's chroniclers got Chalaque, a word used to refer to people they encountered in the mountains of North Carolina, who might have been the ancestors of those English colonists later recognized as the Cherokee.

The Iroquois also have a tangled history when it comes to their names. Iroquois itself is of ambiguous origins, though the common theory has it come to English from French, to French from Algonquin, to Algonquin from Basque. Prior to French colonization of the St. Lawrence Valley, a Basque-Algonquian pidgin had developed to facilitate trade between Basque fishermen and the people of what is now maritime Canada. This is how the Basque hilokoa ("Killer People") could have been converted to Iroquois. Alternatively, it may have come from a Huron (also a name imported from the French; the Huron called themselves the Wendat and after their merger with the Petun / Tionontati as the Wyandot) word irinakhoiw, another term that calls up snake-like insults without the same foreign-language-speaker connotation. The Iroquoian phrase hiro kone ("I have spoken") which once punctuated the end of speeches has also been argued as the origins of the name, but I find that the least likely of the three options here. In English, Haudenosaunee is generally seen as a better term for the Iroquois. It's derived from the word Rotinonshon:ni, which means "They Build A Longhouse," a reference to the common political metaphor of the Iroquois Confederacy as a longhouse extended over and uniting the people. Keep in mind, however, that this is just one name for the League and its members spoke six related but distinct languages.

For the six nations that eventually made up the Haudenosaunee, its a mixed bag when it comes to whether their English names are in anyway related to their own language. The people that are referred to in English as the Seneca and the Onondaga have very similar sounding names in their respective languages: Onöndowága and Onöñda’gega’ respectively. English adopted Onondaga to refer to the Onöñda’gega’, while it adopted a variation on the Seneca capital, Osininka, as the name for the Seneca people. On the other end of the spectrum, the Mohawk refer to themselves as Kanien'kehá:ka; "Mohawk" seems to have come into English by way of the Dutch, who picked it up from the Mahicans who lived between the Dutch in New Amsterdam and the Kanien'kehá:ka. Whether Mohawk is derived from the Mahican for "Man-eaters" or "People from the Bear Lands" is debatable.

I mentioned the Ojibwe and the Odawa earlier, so I should give you an idea of where those names come from. Together the Ojibwe and the Odawa are known as the Anishinaabe, imported directly from their word anishinaabe, which means people, or more specifically Native people, or more specifically still refers only to those Native peoples who migrated from maritime Canada / New England up the St. Lawrence River and into the western Great Lakes sometime within the last millennium. This last group includes the Ojibwe, Odawa and Potawatomi as members of the Council of Three Fires, but also includes a few off-shoots like the Algonquins, Nipissing, and Mississaugas, which branched off from the main migration at various points in history. The Ojibwe, Odawa, and Potawatomi speak dialects of the same language; for simplicity I'll use the Ojibwe dialect here. The names for the three divisions of their council come from the roles each division was to play. Potawatomi is related to the word boodawe ("s/he builds a fire"); they were the Firekeepers, responsible for hosting and moderating council meetings. Odawa is related to adaawe ("s/he buys") and were skillful and far-ranging traders. Ojibwe is related to ozhibii'ige ("s/he writes"); they were the keepers of the faith, which included maintaining the wiigwaasabakoon--the sacred birchbark scrolls. From Ojibwe and a common habit of dropping initial vowels and turning J's to Ch's and switching B's and P's back and forth, English also gets the word Chippewa. EDIT: I forgot to mention that linguistic origins of their names aside, the Potawatomi, Odawa, and Ojibwe didn't (and especially now don't) have a monopoly on those aspects of societies.

Usually they were named by the neighboring tribe that the European encountered. Sioux in another language means snake, they do call themselves Sioux now since its the only name most people recognize. The names were either descriptions of the people, where they lived, or they would be insults depending on how well the two got along. For example there's the Winnebago, who were named by their neighbors "Stinking Water". There are a few tribes that had their own names, but having a name for your particular group wasn't as common as you'd think. Today some tribes will just use their word for people as a proper name for themselves.