Slavery was popular in the United States in the 19th century for two main reasons. First, it was popular because it was (or at least it seemed to be) economically beneficial to the country. Second, it was popular because whites of the time generally held very racist beliefs.
One reason for slavery was economic. The South had built its agricultural economy on slavery and it would have had a very hard time changing the system. All of the planters in the South had huge amounts of capital tied up in their slaves and would have lost all of that money if the institution had been abolished. The North had no slaves by the middle of the 19th century, but its economy was built at least partly on slavery as well because it sold goods and services to the slaveholders of the South.
A second reason for slavery was racism. Slavery was popular because most whites had very little trouble with the idea of enslaving black people. Many Southerners believed that slavery was a positive good for the slaves because it was better (in their view) than living in Africa or than working for wages in Northern factories. Even people in the North who did not think that slavery was necessarily good did not really care about the rights of black people very much. They felt strongly that blacks were inferior to whites and so they did not care what happened to them.
Slavery, then, was popular because it was economically beneficial to the United States and because, due to racism, most white Americans did not have any moral qualms about enslaving blacks.