That was the old way of referring to slavery. The word "peculiar" today means strange or odd, but in the 1800s, it meant uniquely one's own. Keeping slaves was unique to the southern United States, and rather than just saying that, people had some polite expressions that did not sound as harsh and covered up how brutal slavery was.
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The "Peculiar Institution" was and remains a common euphemism for slavery in the U.S. southern slave states. People to this day will speak of "the South's Peculiar Institution" as a way of referring to slavery without actually using the word "slavery."
Our) peculiar institution
i don't know about the following but a well known euphemism for slavery is the "peculiar institution"
"The peculiar institution" set the South apart from the rest of the country.
That would be an adjective.