Non-Newtonian Fluids are materials that act (not surprisingly) in a Non-Newtonian manner. For anyone who isn’t into dense physics, basically this means that as you increase the speed and force with which you try to pull, push or tear a material, you change the strength of the material. That probably doesn’t make sense either, but think of silly puddy. You know how if you pull on it a little bit and let it hang in the air, it will slowly droop and stretch out? Now if you take that same silly puddy and try to pull it really fast, it will either tear quickly or resist the motion. Silly puddy is a type of Non-Newtonian fluid called a Dilatant, which gets stronger the faster you try to move it. On the other hand, something like water should act the same whether you act on it fast or slow. And there is another type of Non-Newtonian fluid called a pseudoplastic, like paint. When you try to roll paint on to a wall, it becomes softer which makes rolling easier. Once the paint is on the wall, it thickens again and will not drip off.
The basic explanation of why Dilatants (like the corn-starch water mixture in the youtube video) act so weird involves polymer chemistry. Cornstarch forms long molecules which can move in water. When they are pulled quickly, the molecules snag up on each other and get tangled, making the solution seem more like a strong solid. When they are pulled slowly, however, they can slide past each other like spaghetti. In the youtube video, the cornstarch is vibrated rapidly and harmonically, bringing out the Dilatant properties.