In a periodic tiling you can take a copy of the original tiling and slide it some combination of left/right and up/down such that it will look exactly like the original tiling. The most basic example is to imagine an infinite sheet of grid paper. If you slide it around you wouldn’t be able to tell whether you are looking at the original or the copy. This is called translational symmetry.

In an Aperiodic tiling you can’t do this. Such filings do not have translational symmetry. You might be able to find sub patterns that look like other sub patterns but if you try to map them to each other you’d find that the tiles around them eventually don’t line up. There are many such tilings.

What’s unique about “the hat” is that it’s the first time an aperiodic tiling has been made using a single shape. The first aperiodic tiling used thousands of different shapes. And before now we’d been able to find multiple aperiodic tilings using as few as two shapes. But this is the first one which uses just a single shape. And to make it even more interesting it was found not by a professional mathematician but a hobbyist (though he did work with two universities to prove the shape he used was, in fact, capable of aperiodic tiling).