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You should use it for times when you don't want to have to breakpoint every little line of code to check variables, but you do want to get some sort of feedback if certain situations are present, for example:
Debug.Assert(someObject != null, "someObject is null! this could totally be a bug!");
Assert allows you to assert a condition (post or pre) applies in your code. It's a way of documenting your intentions and having the debugger inform you with a dialog if your intention is not met.
Unlike a breakpoint, the Assert goes with your code and can be used to add additional detail about your intention.