To deprecate is to show disapproval or to make someone feel unimportant by speaking to them disrespectfully, like seniors who deprecate younger students just for fun.
To deprecate is to diminish, or to oppose, like when someone deprecates your dream of climbing Mt. Everest by calling it "a little walk up a hill." Some people deprecate themselves, which is called being self-deprecating. People who are self-deprecating downplay their abilities because they are humble — or want to appear that way. So if you do climb Everest and you are self-deprecating, you'll say it wasn't such a big deal, even if it was the proudest moment of your life.