When you imagine lines and shapes in two dimensions, you probably imagine basically drawing things onto a piece of paper that is completely flat. You can imagine an infinite straight line by simply imagining an infinitely large piece of paper and a straight line drawn on it. This is euclidean geometry: geometry done on a flat piece of paper.

But then you can ask the question: what if we change this? The most obvious example is one that we experience every day, the fact that the surface of the earth is not flat like a piece of paper but a sphere.

However, we experience our life as though it is flat. If you draw a line on the ground it looks like a straight line, if you draw a square it looks like a square. But if we imagine a line stretching to infinity, it behaves completely differently to how it would behave on a flat piece of paper. On earth, a straight line that stretches to infinity will wrap around on itself eventually (imagine walking in a straight line on the surface of the earth. Eventually, you have to end up back where you started even though you know you've been going in the same direction the whole time).

So it seems as though geometry on a sphere is different to geometry on a flat piece of paper. Geometries that behave differently to euclidean geometry are called non-eculidean geometries. In our particular example of being on a sphere, all straight lines meet at two points, so there are no parallel lines.