Slavery is a system under which people are treated as property to be bought and sold, and are forced to perform various kinds of work. Slaves are typically held against their will. People can become slaves when they are captured as prisoners in war or raids, and some are even sold into slavery by their parents if they are poor and cannot afford them. Others are born into slavery, as the children of other slaves. Slavery has officially been outlawed in the continent of Westeros for thousands of years, but it is widespread throughout much of the eastern continent of Essos.
After Daenerys Targaryen succeeded in liberating Slaver's Bay under her rule, the slave trade has functionally been abolished throughout Essos (though slaves are still owned by masters in slave cities as can be seen when Tyrion Lannister and Varys visit Volantis in Season 5). However, it remains uncertain how long this abolition will last since Daenerys has left for Westeros, leaving the cities of Slaver's Bay without her rule, having Daario Naharis as her regent in her absense.
Slavery has been illegal in the Seven Kingdoms for thousands of years. Slavery is considered an abomination by both the Faith of the Seven and the worshipers of the Old Gods of the Forest. It was illegal in Westeros long before there even were "Seven Kingdoms", and the unification of the Seven Kingdoms in the Targaryen Conquest three hundred years ago only served to strengthen the enforcement of this prohibition.
Even selling criminals to foreign slavers from Essos is considered a heinous crime. When Lord Jorah Mormont sold some poachers he caught on his lands to foreign slavers, he fled into exile to avoid execution by his liege Lord, Eddard Stark.
The ironborn push the limits of this with their practice of taking thralls, but they are technically not slaves, as even the ironborn raiders don't believe in "slavery" as such. Thralls are captured in raids, by "paying the iron price", but cannot be bought or sold like animals. The ironborn generally look down on selling slaves, as they still think of their captives as human beings, albeit on the bottom rung of society. Thralls are more like "prisoners of war" bound into forced servitude or labor. The children of thralls born on the Iron Islands are free men and are even considered ironborn if they are baptized under the faith of the Drowned God.
In sharp contrast with Westeros, slavery is widespread throughout the continent of Essos, across the Narrow Sea. Large-scale chattel slavery has been practiced in Essos for thousands of years, since the dawn of recorded civilization. Indeed, it would be simpler to list the few places in Essos where slavery is not practiced: the Free City of Braavos, one or two of the other northern Free Cities such as Pentos which Braavos has managed to force to abolish slavery (albeit only nominally) and Lhazar. The slave trade is so pervasive in most of Essos that in many ways, slaves are used as a form of currency. Indeed, the slave trade is one of the pillars of the international economy in Essos.
Cultures from Essos frequently make slaving raids on the neighboring continent Sothoryos, to the south across the Summer Sea. Naath, a large island off the west coast of Sothoryos, is also a frequent target of slaver raids. The large island chain of the Summer Islands, to the west of Naath and mainland Sothoryos, is home to a powerful local civilization which abhors slavery. Summer Islanders have often warred with slavers in the past and their fleets drive slaver ships away from their waters. Slavers still make some opportunistic raids against Summer Islander holdings when they can, however, so it is not unusual to see Summer Islander slaves in Essos.
Valyrian Freehold and Ghiscari Empire
The ancient empires of the Valyrian Freehold and Ghiscari Empire practiced slavery since the beginning of their histories. The Ghiscari Empire was built on the backs of slaves, but it was conquered by the Valyrians and their dragons five thousand years ago. The Valyrian Freehold went on to conquer half of the known world, and their domains in Essos were fueled by enslaved peoples. Slaves worked the fields and died by the tens of thousands toiling in their mines. After the Doom of Valyria four hundred years ago, slavery continued in the surviving Valyrian colonies which became the Free Cities and Slaver’s Bay.
Slaver's Bay is the central hub of the international slave trade. Qarth is also involved in the slave trade, and Asshai seems to be as well. The cities of Slaver’s Bay breed their local slave population, and also capture new slaves in raids on distant shores, such as the coasts of nearby Sothoryos (including the island of Naath). They trade their slaves to the southern Free Cities for food and finished goods, as well as to eastern locales such as Qarth and Asshai for more exotic products and spices.
Slaver's Bay is also known for its fighting pits, where slaves are forced to fight to the death with each other or against wild animals.
Many of the Free Cities also practice slavery, and the southern Free Cities in particular have large slave populations. Slavery is practiced in Volantis, Tyrosh, Lys, and Myr. As Volantis is most southeasterly of the Free Cities and the closest to Slaver's Bay, it is the main hub of the slave trade in that part of the continent. Due to the dangers of passing the ruins of Old Valyria, ships going to or from Slaver's Bay will usually stop at Volantis first. As a result, while Volantis is the most populous of the Free Cities, its population contains five slaves for every one free man. Lys, known for its pleasure houses, also holds a significant slave population.
The slave-masters in Volantis have a very organized system for their slaves, branding them with small tattoos on the left cheek of their faces that symbolize what tasks they are assigned to: flies for dung shovelers, hammers for builders, tears for whores, etc. Thus even freedmen remain marked. Slaves are also forbidden to touch nobles, under penalty of death.
Braavos, the most powerful of the Free Cities, is the major exception; it was founded by slaves who fled Valyria, and to this day the Braavosi abhor slavery. Slaves brought to Braavos, such as on a slaver ship that was blown off course, will be forcibly freed if they land in Braavosi territory. Pentos was forced to nominally give up slavery in a treaty made with Braavos a century ago, but many of its "servants" are actually slaves in all but name. Some of these servants are even forced to wear heavy bronze collars.
As for the other northern Free Cities, Qohor and Norvos do practice slavery, but not nearly on the large scale in southern cities such as Lys or Myr. Lorath apparently doesn't practice slavery either.
The Free Cities frequently pay tribute to the Dothraki hordes in the form of slaves, as this is less destructive than suffering a raid.
The Lhazareen are another exception that does not practice slavery: they are instead frequently the targets of slaving raids by the Dothraki. The Dothraki capture slaves from lands surrounding the Dothraki Sea, such as Lhazar and other lands much further afield (combined with the slave tributes from the Free Cities).
The Dothraki then give slaves to Slaver’s Bay or Qarth in exchange for weapons, finished goods, and luxury items, completing the cycle. Theoretically, a slave born in Slaver’s Bay could be traded to the Free Cities in exchange for weapons, then gifted as tribute to the Dothraki, and finally traded by the Dothraki back to Slaver’s Bay, in exchange for the very finished weapons from the Free Cities that they were traded for in the first place.