Good question! Well first let's look at the periodic table and how it works.

The periodic table contains all of the elements we have ever had or created on earth. Elements after 95 (and also elements 43 and 61) are not naturally found on earth and have to be made in the lab. If you've forgotten what elements are then check out my post on them!

Well, there are a few different ways actually, but I'll stick to the ones you need to know for Chemistry. The first one is diffusion. Scientists back in the day observed people smoking and thought why does the smoke disappear? Then they thought well for it to move there must be something pushing it (Newtons 2nd Law!) and the only reasonable culprit was atoms. This was backed up when a Mr. Brown was looking at pollen grains in water under a microscope when he realised that the pollen grains were jiggling around. He called this motion Brownian motion after himself like they all do. Then later on the genius Albert Einstein came along and made the link between Brown's jiggling pollen grains, diffusion and atoms and there it was. Since then our idea of what atoms are has changed, from the plum pudding model, to the nuclear model and finally to the current quantum mechanical model of the atom. For GCSE forget about the quantum mechanical atom and focus on the nuclear model. This is the 'right' one.