A periodic tiling of the plane is one on which you can outline a region that tiles the plane by translation. In other words, if the plane is divided into tiles, then the tiling is periodic if you can slide the plane onto itself by a translation so that the tiles of the original copy fit exactly the tiles of the new copy.
(1) There are many shapes that tile the plane only periodically, and there are other shapes that tile the plane both in a periodic and a non-periodic fashion.
The question then arises as to whether there exists a finite collection of shapes that can tile the plane but only in a non-periodic fashion.
In 1961, H. Wang conjectured that given a finite set of tiles, there is an algorithm to decide if that set of tiles could be used to tile the plane. He also showed that this would be true if every set of tiles which tiles the plane also tiles the plane periodically periodically. His tiles (Wang dominoes) were unit squares with edges colored in different ways, and tilings must be constructed by placing then side by side so that colors match. The problem is important for it relates to decision questions in logic (Turing machines).
R. Berger solved Wang's domino problem in the negative by producing a set of more than 20,000 tiles which tile the plane, but only non-periodically. Later the number was reduced, the smallest known set was produced by R. Robinson.
A few years later, R. Penrose found a smaller set of tiles which tile the plane only in a non-periodic fashion. His set consist of two tiles. Their shape can vary; the most popular are perhaps the `kite and dart' shapes.
(2) Describe the (several) set(s) of Penrose tiles.
(3) Show that they indeed tile the plane and only in a non-periodic fashion.
There are many facts known about the Penrose tiles. The references below contain plenty of material about this topic.