What events and forces prompted the expansion of American slavery between 1790-1820?How would you describe the slave trade?
This is an interesting, if depressing, question. Just so that you know, the second part of your question, describing the slave trade, is probably better left asked as a separate question on eNotes because it is a different ball of wax and requires its own detailed answer. I'll focus on the first part you asked.
The expansion of slavery in the US during this time period is a reflection on the growing state of Southern agriculture. As the amount of land under cultivation grew, so was the need to have more labor to do the work. So, at its base level, that's a factor in itself; as slavery spreads to new states and territories, more slaves are needed, and as more land is put under cultivation in existing states, the same thing happens.
Another factor was the type of crops being grown. Tobacco was the big money maker for a long time. Don't get me wrong, it's no fun to go out and work in a tobacco field, but it's similar to harvesting many other types of crop; cut the plant down, dry it, sell it. With the invention of the legendary "cotton gin," though, during this period it became profitable to grow and sell cotton. Cotton is a different animal and a lot bigger pain in the you-know-where. Here is a picture of cotton being picked around the time of the Civil War. It's not the period you're asking about, but the idea is the same: LINK.
So that's the biggest chunk of your question. Because of the cotton gin (and a few other newfangled precious-es, such as the Spinning Jinny,) cotton became a crop that could be grown economically and sold for big money. The drawback was that it's hard work and people had to go out into the field and pull of each little "puff" of cotton by hand. The places cotton grows well are not pleasant places to be working during summer in the South, as you can imagine. Forcing people to do the work for free seemed like a logical, albeit horrific, way to do business.