/u/CaptEdmundBlackadder's answer is essentially right and on the mark, but I would expand on his answer a little bit.

As /u/CaptEdmundBlackadder alludes to, there isn't any way for us to neatly pinpoint an origin point of the transatlantic slave trade. The European use of African slave labour has its origins in the mid-1400s and most probably began with the enslavement of Arab and African people during the course of the Reconquista, but this was not yet an organised slave trade organised along clearly racialised lines - this is the first time in our history that Europeans began to meaningfully think about the idea of 'race', and the evolution of the idea of 'race' in terms of black and white is one that grows up alongside slavery, both informed by and informing its development. Portugese expeditions into Africa began in the early 15th century and most probably started bringing enslaved Africans back to Portugal with them then. In 1452 Pope Nicholas V issued a papal bull titled Dum Diversas that sanctioned, for Portugal specifically, the enslavement of non-Christians and particularly Muslims and although its scope was officially without limit, the context of the bull is understood to represent the sanctioning of the enslavement of people in West Africa.

But Portugal very quickly found that there was a more effective way to engage in the slave trade: by connecting with local, existing mercantile networks and existing systems of forced labour, which existed in some diversity across the region (although not always in the form of true slavery, and the scale of their practice has been debated). A subsequent bull by Nicholas V in 1454/5 reaffirmed Papal sanction of Portugese military activity in West Africa and further sanctioned the enslavement of African people through trade and not simply war, ostensibly on the rationale that through enslavement it would be possible to Christianise the population more rapidly. Many thousands of men, women and children were carried into bondage out of Africa and into Portugal itself during this period of the slave trade, and this is where the slave trade truly begins - but it is a period for which records are scarce and tracking the development of the trade is extremely difficult. The infrastructure that would facilitate the later trade in enslaved people across the Atlantic ocean into the New World can probably best be said to have emerged by 1482 when Elmina Castle, or Castelo de Sao Jorge da Mina as it was known at the time, became operational in the early 1480s. Elmina Castle was essentially the first processing and logistical hub for the West African trade; a place enslaved people could be taken and imprisoned whilst awaiting transport to Europe, and to which local slave traders could bring their victims for sale to the Portugese as well. The vast majority of people who were sold into slavery to Europeans in this way were people, particularly men, who had been taken as prisoners of war in conflicts between West African states or were otherwise convicts of some kind; whilst this in no way diminishes the inhumanity and injustice with which they were treated, it is important to emphasise that this was the framework through which West Africans complicit in this fledgling trade in slaves to Europe were working, and that their perception of the slave trade was fundamentally different to that of European slavers. In the long-term the hunger for slave labour in the Americas, combined with European military and economic power, would have an unfathomably destructive effect on West African societies as some elites increasingly saw conflict and harsh systems of public justice as a way of achieving economic advantage or prosperity through the slave trade.

European colonisation of the Americas began earnestly in 1494 when Christopher Columbus founded the town of La Isabela on Hispaniola, in the modern-day Dominican Republic, and from there other colonial enterprises followed. When exactly the transport of enslaved people to the New World rather than Europe began is unclear thanks to the partiality of contemporary records. Early European colonial projects tried to make use of Native American slave labour instead of West African forced labour which would have not immediately seemed as practical or expedient in the cold economic calculations of European powers given that vast distance and dangers inherent in crossing the Atlantic ocean. We think that the first enslaved person to be taken to the Americas probably left in 1502 as the personal slave of a Spanish merchant named Juan de Cordoba (not to be confused with the priest of the same name and home town) though it is possible someone else beat him to that unpleasant record. Spain authorised the limited export of enslaved people to its New World colonies by royal decree in 1510, with an order to expand this miniscule trade given in 1518. The Monumenta Missionaria Africana, a collection of documents relating to missionary activity in West Africa by the Roman Catholic Church that also touches on details of European activity in other ways, gives us the first known description of a 'true' slave ship in the year 1514, but its destination is not known. The first slave ship we know the definitive details of set sail in 1525: the San Maria de Bogona, a Portugese ship that delivered an estimated 250 - 300 souls we think to Hispaniola, but by then several thousand people had already been taken into slavery in the Americas by the Portugese and Spanish. It is from 1518/19 that we begin to see the first real, organised demand for slave labour on a meaningful scale, and we can probably say that this represents the beginning of the transatlantic slave trade in earnest.

Finally, to expand on /u/CaptEdmundBlackadder's point about the end of the transatlantic trade, the slave trade to Brazil was notionally abolished in 1850 - but it continued to Cuba until the very end of the US Civil War. The last slave ship we have surviving records for is a Portugese vessel named the Daome, intercepted by the British attempting to carry enslaved people to Cuba in mid 1866, although we know the last slave ship arrived in Cuba itself in 1867 and lack detailed information about it (the Daome, for its part, was caught in Africa before it could embark any victims). Slavery itself persisted in Cuba until 1886. As pointed out by /u/CaptEdmundBlackadder there was quite possibly some slaving activity here and there until the end of slavery in Cuba and Brazil that went unnoticed - we know several thousand enslaved people were still arriving in the United States for years after the slave trade was abolished there and that there was some small-scale fragmented illegal trading to the British Caribbean right up until abolition. By and large however historians consider the transatlantic slave trade to be in its final days by the mid-1850s; in 1855, the number of annual victims of the slave trade fell below five digits for the first time since 1648 and though it had a brief surge of recovery in the following years, it is functionally extinct by 1867.

All in all we estimated that there were about 36,100 Atlantic crossings - roughly 100 every year between 1500 and 1866 - made by slave vessels, each carrying an average of 350 men, women and children into bondage. At its peak between January 1st 1820 and December 31st 1829, a person was taken against their will from the West African cost on average every 1.6 minutes, every single day, every single week, every single month, every single year, without interruption for the entire decade. We know the details of the ship in about 32,000 of these cases, and details of one or more of the financial sponsors of the trip in about 20,000, but in only fewer than 10,000 cases do we have details about the African victims of the trade in any form - mainly thanks to the more rigorous efforts of the British to monitor the transatlantic trade in terms of the number of victims, both during the British Empire's period as a slaving power and later as an interdicting force against slavery. Only around 91,000 African people are known to us by name and specific details of their lives or health - about 0.73% of the total number of people taken by European slavers during the trade's history - and these records come from efforts to impound slave ships by the British after the abolition of slavery. By and large European slavers were not interested in recording the details of their victims, even necessarily their exact total number, beyond being able to comply with local laws and maximise their profits, and so the records we have from the trade are often lacking in the kind of details historians would be most interested in recovering.

As an aside, special mention should be given to the journey of 1563 made by three British ships under the stewardship of John Hawkins, a major figure in the history of the Royal Navy. Hawkins took three ships in 1563 - the Solomon, Swallow and Jonas - on the first recorded British slave trading voyage, and he would repeat it multiple times in the next few years. This is often taken to be the first true voyage of the 'triangular trade' that made transatlantic slavery so profitable not only as a means for getting forced labour to the New World but as a merchant enterprise in and of itself and, whilst the Portugese and Spanish were already doing something similar and created the original triangular trade with the Canary Islands off the coast of Africa, it was the British Empire's later forays into slave trading with her North American colonies that made slavery into such a hyper-profitable global enterprise.

It seems that slaves were first transported across the Atlantic sometime in the early 16th Century, but in very small numbers; in the early years of European (primarily Portuguese and Spanish) colonisation of the Americas, experiments in using local, native American slaves were made. These first African slaves were likely domestic slaves of merchants, nobles and the like. Groups of maybe 20 slaves at most may have been transported, but not as part of any 'trade' in the sense we understand today.

In 1517 a licence was granted to Jorge de Portugal to import 400 black slaves in the Caribbean, but it seems the licence was never executed. In 1518, the Spanish on Hispaniola (modern day Haiti/Dominican Republic) requested black slaves to replace the indigenous population they'd already managed to work to death. As a result, a licence for 4,000 was granted.

It therefore seems - and the records are either lost or very limited - that the 'first transatlantic slave ship' began its voyage in late 1518 or early 1519. By the early 1520s ships were regularly transporting around 200 slaves at a time, this figure ultimately reach as high as 1,000 in the 18th Century.

The last slaver is much harder to establish, as the trade continued well after abolition began to roll out across the world. The Royal Navy's West Africa Squadron ('The Preventative Squadron'), which existed from 1808-1867, seized some 1,600 slave ships during its existence, but when one considers that Brazil did not abolish slavery until 1888, a trade almost certainly continued until around that time.

However, the last legal British slaver was Kitty's Amelia, which was cleared to sail on 27th April 1807. Abolition came into effect on 1st May of that year. One could argue that as the Royal Navy would effectively claim international jurisdiction to eradicate the slave trade, Kitty's Amelia was the last legal slave ship.

The last known US ship to bring slaves to America was the Clotilda, in the summer of 1860. The subsequent Civil War put an end to any further voyages.