I am not a chemist, but I am interested in Science in a general sense. Can anybody explain why the periodic table is periodic in nature? I would appreciate links for further reading.
Elements interact with the rest of the world through their electrons. How those electrons interact with other atoms (or electromagnetic radiation) determines how that atom behaves.
You see this with isotopes. Different elements are distinguished from each other by the number of protons in their nucleus. Changing the number of neutrons has little effect on their properties. Why is this? It's not the protons themselves which change the properties, instead it's how the change in total positive charge of the nucleus changes how many electrons the atom can hold and how they behave that changes the properties. Different numbers of electrons - and specifically different numbers of electrons with respect to the net charge on the atom - change the properties of the atom. Changing the number of neutrons changes the nucleus, but has little effect on how the electrons behave, so has little effect on the chemistry of the atom.
So what does that have to do with "periodicity" of the periodic table? Well, there's a structure to how the electrons are "stored" in an atom. These are orbitals. For a bunch of quantum mechanical reasons that are rather advanced for this question, there are repeating patterns in these orbitals. Most particularly, in the l quantum number which determines the shape of the orbital. Each period corresponds (roughly) to the n quantum number. For each n, there are a set of ls which go along. (Because of tradition, orbitals with different l quantum numbers get letter labels: s, p, d, f, ...)
This is why you see periodic trends. Having exactly filled s and p orbitals is the most stable, and each n has its own set of s and p. So each time you get a filled s and p, you get an inert noble gas. For halogens (like chlorine and bromine), you need just one extra electron to fill the s and p orbitals, so they really like taking a single electron from other elements. In contrast, the alkali metals (like sodium and potassium) can most easily get to a completely filled s and p orbital state by losing a single electron, resulting in their propensity for a positive charge. It's how close the atoms are to filled s and p orbitals - regardless of their n quantum number - and what it takes to get there which provides the bulk of the "periodicity" of the periodic table.
It turns out that the s and p orbitals are the most "exposed", so what's going on there contributes the majority of the behavioral change between elements. Elements that differ only in how the electrons in their d orbitals are situated have much less property differences (this is why most transition metals have similar properties). Elements which differ only in f orbitals (like the Lanthanides) are even more alike. There certainly are differences here, but they tend to rely on more subtle effects than those that cause the alkali/chalcogen/halogen/noble gas differences.
Note that this sort of orbital-based periodicity breaks down as you get lower in the periodic table. That's because you actually start to get relativistic effects, and how the orbitals behave starts to change. Periodic trends are just that - trends - and aren't absolute.
The answer can be really simple indeed. The elements are organized according to their atomic number in ascending order, and if you analize the properties of the elements in a period (row) or in a group (column) you'll see that the properties of those elements follow a "pattern". For example generally speaking the atomic radius in a group increases with the increase of the atomic number and insted ionization energy decreases. This has exceptions, of course, but this regularity helps to have a fast idea about the properties of an element