The distinction between a "society with slaves" and a "slave society" was first made by scholars of the ancient world. When it was adapted to the context of the Atlantic World, historians like Ira Berlin have used it to describe the difference between societies like colonial New York, where slavery was widespread but not central to the economy or to social structures, and the Carolina Low Country and the Caribbean, which were entirely built around the institution. At least in the colonial period, slavery and unfree labor were important almost everywhere. But in "slave societies," the laws, the economy, and even politics were structured around the institution of slavery.
Usually the establishment of a slave society occurred in response to economic forces. For example, in the years following the founding of Jamestown, the Chesapeake emerged as a slave society due to the importance of tobacco as a cash crop. Similarly, the profitability of cultivating rice in the Low Country, and sugar in the Caribbean, led to the establishment of enormous populations of enslaved people there.
Percentages are one way to think about the difference between a "slave society" and a society with slaves," but this model can be problematic in the British American colonies, because slave populations were far more dense in the Tidewater and coastal regions (where cash crops were cultivated) than in the Piedmont and mountains. By the outbreak of the American Revolution, over 50% of the population of South Carolina was enslaved, but this number does not accurately reflect realities. In the upcountry, there were relatively fewer slaves so we could describe this as a "society with slaves." Some counties in the Low Country, however, were more than 98% enslaved. In the Atlantic World (unlike antiquity), the fact that slavery was tied to race meant that a "slave society" had, as historian Peter Wood described it in a book by that name, a "black majority."