I've read through the entire Internet world and I feel even more confused than before. What's the difference between one and the other? What's a for and against essay then? Is it an argumentative essay?
I'm not familiar with either term specifically, but I would guess that an opinion essay might only have to include your personal opinion on the subject, which presumably only gives one viewpoint. In an argumentative essay, I guess you are expected to show two (or more) sides of an argument and then perhaps come to a conclusion at the end. That would be the same for a "for and against" essay. A "for and against" essay about bullfighting, for example, would give both the view supporting it and also arguments for it to be banned.