The question is pretty much in the title, but I am really seeking countries which kept slaves around the same time the US did, and where it was just as prominent in said country as it was in the US.
Most African slaves were sent to the Americas — basically a new world with large industrial-scale labor-intensive agriculture. There isn't much point in taking slaves to Europe to replace serfs or cheap farm laborers on small farms.
Destinations of slaves Africa→Americas
Portuguese America (modern Brazil) 38.5% British America (minus North America) 18.4% Spanish Empire 17.5% French Americas 13.6% British North America 6.45% English Americas 3.25% Dutch West Indies 2.0% Danish West Indies 0.3%
Integrated over 10,000 years of history there were probably far more slaves in Africa from Africa, but the numbers are harder to come by.
Various Middle Eastern countries (e.g. Omani Empire which had the eastern coast of Africa) participated in the East African Slave Trade (a.k.a. Arab Slave Trade).
Slavery in Mauritania has long been an integral part of society in this West African country. It dates back centuries and continues to this day (so this more than covers the time when there were slaves in the US). It is estimated that there were still well over 100,000 slaves in Mauritania in 2014, and the number could be several times higher than this. Slavery was only made a crime in 2007 but enforcement of the law has generally been weak.
Slaves were sold to Europeans from Mauritania during the Atlantic slave trade, but hundreds of thousands were also enslaved locally before, during and since this period. A key event in the history of slavery in Mauritania was the Char Bouba war (1644-74). This resulted in
...putting Arabs in control, with Berbers in the middle (divided into 2 groups: religious scholars called zawiyas and farmers and herders called znaga), and the blacks on the bottom (also divided into 2 groups: former slaves called haratani and slaves, abid).
The French outlawed slavery in 1905 but were unable (and, ultimately, unwilling) to effectively enforce the ban without alienating the local elite:
During the colonial era, slavery in Mauritania was presented as "different" from slavery elsewhere in French territories. At the time of conquest, it was argued that recognition of the "morals, customs, property and religion" of the Muslim Mauritanian elite necessitated permitting it continued access to slaves....it was felt that precisely because slavery was so deeply rooted, slaves themselves were simply "not ready" to be wrenched from their social security, to do so would be to "sow social disorder".
Source: E. Ann McDougall, 'Living the Legacy of Slavery'. In Cahiers d’Études africaines, 2005.
It was abolished again in 1981 by the Mauritanian government but enforcement has been weak to say the least.
Slavery in Mauritania is hardly ‘modern.’ It is an institution deeply rooted in the history of the country and region. The ruling minority Beydanes (Arab-Berbers) historically enslaved Haratin (or “Black Moors”).
No one knows for sure how many people are enslaved in Mauritania today but a conservative estimate is 43,000 (1% of the population). In 2014 - more than 30 years after abolition - the number was estimated to be at least 140,000 (3.5% of the population), but other estimates put it as high as 680,000 (20% of the population).
Despite the indisputable evidence of slavery in Mauritania, the government has long denied it. Witness this statement by the Mauritanian representative to the United Nations Committee on the Rights of the Child:
Mauritanian society had never known servitude, exclusion or discrimination, either in the pre-colonial or colonial period or since independence, and so no vestiges of such practices could thus persist (Amnesty International 2002, p.29)
Cited in: A. Abou Toure, 'Slavery in Mauritania' (MA thesis, 2012)
As Abou Toure notes though,
Slavery is both found in the rural areas, where it takes a more traditional form, and in the urban settings in Mauritania. These slaves are found doing backbreaking work on every street, every field, and are tending to goats, sheep, and camels in the hot desert or semi-arid regions of the country.
Slavery was also widespread in parts of West Africa (e.g. Mali, Ivory Coast, Burkina Faso) up until the late 19th century. Only the defeat of the Malinke ruler Samori Toure in 1898 allowed the French to effectively put an end to the practice.