Hi Darby, that's a great question.
To oppress someone or a group of people means to actively persecute them or take measures to keep them down. For example, "The country's leader had a history of oppressing any groups of people who disagreed with his government."
To suppress something means to silence it or to keep it down, but it's more commonly used for things like ideas, voices, rights. You wouldn't usually say you're suppressing a person, but you might say you're suppressing their voice. For example, "The country's leader had a history of suppressing the voices of any groups that disagreed with him." "Suppress" is also commonly used to talk about a person being unhealthy because they stuff their emotions. For example, "When I was growing up, we weren't allowed to show anger, so whenever I felt angry I learned to suppress it."
Instead of "dominant," I think maybe you meant "dominate"? (But if dominant is what you meant, let me know and I'll fix my explanation :) To "dominate" means to exercise control over someone or something (such as animals). An abusive husband might dominate his wife. An alpha male dog would dominate all of the other dogs in the pack. It can also be used in the world of sports. If a particular player or team is performing exceptionally well, we might say that they are "dominating" all of the other players or teams.
Hope that helps!