A khalasar is a Dothraki clan or tribe. Khalasars can be quite small but can expand to encompass tens of thousands of mounted warriors and many more than that number in women and children. Khalasars are spread across the Dothraki sea and frequently raid surrounding regions.
Khalasars which meet on the plains usually engage in battle, except at Vaes Dothrak where no weaponry is permitted due to the illegality of bloodshed within the city. However, that does not mean that violence in the city is unheard of, with Dothraki devising cunning means to kill without the use of steel or shedding of blood.
Khalasars sometimes differentiate themselves by decorating their bodies and their horses with different colors of paint. For example, Khal Drogo's khalasar used blue paint, but Khal Jhaqo's khalasar uses red paint. Such dyes are expensive so they tend to only use them for special occasions.
In the books
Khalasars are subdivided into khas, each of which is commanded by a captain known as a ko (plural, kos).
Leadership of a khalasar is not strictly hereditary. Often when a khal dies his khalasar will break up into new smaller ones led by his surviving kos. The Dothraki follow strength, so if the son of the current khal is weak or underaged, they have no reason to follow him (i.e. Daenerys carrying Drogo's unborn son was no reason for his khalasar to continue to follow her). This way the Dothraki never have a weak child leader inherit rule, which might imperil their survival (such as when the eight year old Tommen Baratheon inherited the Iron Throne in Westeros). Other times, however, the son and heir of an older and successful khal may have himself grown to young adulthood and become a strong and revered warrior in his own right - in which case leadership of the entire khalasar will indeed pass to the son of the khal, instead of breaking up upon his death. In such cases the adult son and heir of the current khal is known as a khalakka: Drogo was khalakka to his father Bharbo and directly succeeded him upon his death, given that by that time Drogo was already a skilled and respected warrior.
Dothraki khalasars have no names, nor do any have particularly long-lasting or deep histories and affiliations. Khalasars frequently dissolve on the death of their khal into new, smaller khalasars. On the other extreme, when one khalasar conquers another khalasar, if often absorbs the remaining forces of the defeated khalasar instead of simply massacring them. In this fashion one khalasar numbering only a few thousand riders can rapidly aggregate more and more lesser khalasars into itself until it is a single large khalasar numbering in the tens of thousands. The actual number of riders in any one khalasar is therefore fluid, as they are frequently dissolving, coalescing into different groups, dissolving again, then re-combining with entirely different groups, etc.
Generally, the khalasars considered large number in the low tens of thousands (the adult male mounted warriors - the women, children, elderly, and slaves moving with the entire khalasar will number more than that). Several khalasars of this size seem to be active at any given time. Khal Drogo's khalasar contained about 40,000 mounted riders, which was considered unusually large. Back during the great wave of Dothraki expansion in the Century of Blood following the fall of Valyria, a few khalasars reached even larger sizes. Khal Temmo's horde at the Battle of Qohor was said to number 50,000. The largest battle of the Dothraki in the Century of Blood was the "Field of Crows", the climactic battle in their campaign against Sarnor, the coalition of city-states at the western end of the Dothraki Sea. Hoping to finally stop the advance of the Dothraki, the Sarnori fielded an allied army of 26,000 horse and 100,000 infantry. To face this force, four separate Dothraki khals allied their four khalasars into a massive force of 80,000 mounted warriors - apparently an unprecedented number, and they succeeded in crushing the Sarnori. The Dothraki politically fragmented after the first few generations of the Century of Blood, however, so that in subsequent centuries Drogo's khalasar of 40,000 was considered atypically large.
After Drogo was rendered catatonic (and soon died), his khalasar broke up into several smaller khalasars led by his former kos. Pono was the first to leave, with 10,000 riders, and he was followed by Jhaqo who left with 20,000 riders (fully half of Drogo's original khalasar). Jorah explains that over the next few days several smaller bands from the remaining 10,000 riders split off and left as well: Jorah doesn't give the names of their leaders or exactly how many follow each of them, but he does say that (together with Pono and Jhaqo) about a "dozen" new khalasars formed out of the breakup of Drogo's old one. Given that the remaining 10,000 riders divided up into ten or so smaller groups, however - admittedly of greater and lesser sizes - these smaller bands probably didn't number more than around one or two thousand each.
On the other hand, in the years since Drogo's death another round of warring between different khalasars across the Dothraki Sea has led to some being destroyed, and others agglomerating into much larger sizes by absorbing defeated ones. By the fifth novel, Khal Jhaqo's force was again in the east near Lhazar and Meereen, and his khalasar encounters Daenaerys when she flies back there on her dragon. Khal Pono, meanwhile, is said to now have a khalasar of 30,000 mounted warriors - meaning he apparently absorbed several defeated khalasars. Pono's force, however, is now said to be at the western end of the Dothraki Sea, near the headwaters of the Selhoryu River (a tributary of the Rhoyne) - south of Qohor and northeast of Volantis. It is feared that Pono's khalasar will soon menace Volantis, though many anticipate that he will simply leave again after being given customary tribute.
According to the TV series official pronunciation guide developed for the cast and crew, "Khalasar" is pronounced "KHAL-uh-ssar".