What do the periods (horizontal rows) and groups (vertical columns) of the periodic table indicate to us about the elements that are categorized within them?

Periods indicate that a new period at a new principal energy level which is filled with electrons. Groups indicate that the elements share similar properties because their shells are filled with the same number of valence electons.

Explanation:

We see that starting at period 2 Li has a  +1 charge , He has +2 charge, B has +3 charge, C has the potential to have a +4 charge or -4 charge (because it can gain or loose electrons) and N has a -3 charge, etc. Groups however, have similar properties and every element in a 1A - 8A has the same number of valence electrons. All except the noble gases in which He only needs 2 electrons to be in a perfect state. This is its octet as it would be a completely different element if it had more than 2 electrons in its shell. An example of the groups sharing similar properties is if we were to look at group 1A. In group 1A we find H, Na, K, etc. These are all highly unstable elements when alone (not in a compound) and they are very reactive. If you were to put Na for instance in water, it would light up and you'd see a flame sitting on top of the water. A scientist could do that with the rest of the elements in the group and they would see this increasing in reactivity as you go down the periods. Francium isn't visible long enough to be able to see this happen though. It's more of that kind of element that scientists prove that it exists, but not a very common element to get to learn much about.