As part of my research as a Phd student in the field of Einstein's General relativity, I developed two different methods of explaining this theory in a non-mathematical way. These two methods were presented in my first two scientific papers.
The first of these two methods I often use in a lecture that I give to introduce a general audience to Einstein's theory of relativity. The presentation relies mainly on real material models (crafted by myself) that gives a very hands-on explanation of the core of Einsteins abstract theory. I lost count somewhere after my 70'th lecture, but I think I have given the lecture about a 120 times. It is generally very well received. Below are links to a couple of pdf files, with some illustrations and information about the lecture. The first one is in swedish, the second one is in english.
Often members of the audience have questions afterwards. Then the second method is great for explaining black holes, big bang theories and much else. It is amazing how much of Einstein's theory that can be explained without any mathematics at all.