Elements are the smallest, simplest things a chemist works with. Everything else is built of elements.
Things smaller than atoms are the province of physics. Chemists deal with atoms. Even when chemists talk about smaller things, like protons and electrons, it is how they affect atoms or interact with atoms that the chemist is concerned about.
The periodic table of the elements is in one sense just a list of all the building blocks that chemists play with. It starts with the smallest atom, hydrogen, and goes on up, adding one proton at a time to get through all of the known elements.
But the table is also periodic. It reflects the fact that as protons are added, their corresponding electrons can only be added into defined energy levels. Most of chemistry deals with the outermost electrons, since those are the ones that interact with other atoms. And every time an energy level is filled, the next electron added becomes a single outer electron, in a higher energy level. That makes the element behave like the one above it in the periodic table, because all of the elements in a column have the same number of outer electrons.
This periodicity allows us to predict the behavior of elements we have yet to discover. Before the elements scandium, gallium, technetium, and germanium were discovered, their properties were predicted because there were “holes” in the periodic table where an element should have been.
Likewise, the element protactinium was predicted before it was discovered. The same goes for the element hafnium.