Thus far in the math/science subjects:
1. Atoms come together in two basic ways. They either come together by sharing electrons or they come together because their opposite charges attract. The first kind of "bonding" is called "covalent bonding." The second is called "ionic bonding."
2. In the previous post, we talked about how, depending on where an atom sits on the periodic table, it may be prone to gain or lose electrons. So sodium (whose symbol is Na) in column 1--along with lithium (Li), potassium (K), and two more--tends to lose its outermost electron and take on a positive one charge. In column 7 far on the right, chlorine (Cl)--along with flurorine (Fl), bromine (Br), iodine (I) and one other--tends to gain one electron and take on a negative one charge.
This is a match made in heaven. When you put these atoms "in solution," in water, the electrons are free to trade up. Sodium and chlorine become ions, positively charged atoms attracted to each other. When they crystalize into a solid, we call the resulting compound, "sodium chloride," otherwise known as table salt.
As you look at the periodic table, the tendency of an atom to gain an electron goes up the further to the right you go and the further to the top you go. This characteristic of an atom is called its electronegativity, its propensity to gain an electron. So fluorine is the atom with the highest electronegativity because it is the atom most likely to grab an electron.
Related to this characteristic is the ionization potential or ionization energy of an atom. The more energy it takes to dislodge an electron, the higher the "ionization energy." So like electronegativity, the ionization energy increases as you move from left to right on the periodic table, being its greatest with the noble gases, which hold their complete set of outershell electrons the most tightly.
3. So elements combine in certain ways to form compounds. When they combine on the basis of their charges, the total charge has to cancel out to zero. So sulfur is in column 6. When it becomes an ion, it gains two electrons and fills its outer shell (it has 6 electrons to begin with, so two more makes 8, the octet rule). So sulfur can take on a +2 charge.
Hydrogen sulfide is thus a compound where the charges of hydrogen offset the charge of sulfur. But the charge of a hydrogen ion is only +1. So we need two hydrogens to offset the charge of every sulfur. The chemical formula for hydrogen sulfide is thus H2S.
4. An ionic compound of this sort, with hydrogen contributing its positive ions, is called an acid. We know acids in real life because strong acids are dangerous and are used to clean things. The hydronium ion (H+), if it is very "concentrated," can eat skin or your throat if you swallow it. There are weaker acids that are actually helpful or enjoyable in food. Orange juice has citric acid in it, for example. Acids tend to have a sour taste.
Two of the most common strong acids are hydrochloric acid (HCl) and sulfuric acid (H2SO4). In the second instance, you can see that 2 hydrogens balance the charge of an SO4-2ion.  The "ph scale" was invented to identify how strong an acid is. Something with a ph of 7 is neutral in strength, like water. But if something has a ph lower than 7, it is an acid.
5. To balance out acids is another type of ionic compound called a base. The earliest sense of a base came from a man named Arrhenius, who identified a base as a substance that has an OH- in its formula. So sodium hydroxide (NaOH) is one of the most famous bases and, like hydrochloric acid, it will eat your throat or eyes if you come into contact with a high concentration of it. The concentration of a base is any number higher than a 7 on the ph scale.
From a more sophisticated perspective, a base is a substance that attracts protons. So ammonia (NH3) has no net charge, but it attracts protons for other reasons. In the Brønsted-Lowry and Lewis approaches to acids and bases, ammonia can also be considered a base. Bases tend to have a bitter taste.
6. We have strayed a little from our discussion of ionic bonding. Ionic bonds, again, are attractions between atoms or groups of atoms based on charge. This kind of bonding applies especially to the elements in the first two columns of the period table (which tend to go positive) and the sixth and seventh columns (which tend to go negative). The ten columns of transition metals in the middle (between column two and three) also often form ions.
But many other atoms bind together by sharing electrons, a kind of bonding called covalent bonding. This is especially true of organic compounds that are essential to life. Such compounds typically involve carbon.  Natural gas and many fuels are based on organic chemicals.
Compounds that are held together in this way are called molecules. An example of a molecule held together by covalent bonding is sugar C12H22O11. The formula indicates that a single molecule of sugar has 12 carbons, 22 hydrogens, and 11 oxygens.
In covalent bonding, you might think of a molecule as having a certain number of "docking stations" for other molecules to "dock" at. Carbon, for example, since it is in column 4, has four "docking stations." That means that four other atoms can link up with a carbon atom. Hydrogen, in column one, only has one "docking station."
Voila, CH4 is a natural. Carbon has four openings, each hydrogen has one opening. One hydrogen links up with each docking station of the carbon and we have a bunch of happy campers. There are a total of 8 electrons docked, four provided by the carbon and four provided by the four hydrogens. The octet rule is satisfied. CH4, by the way, is methane gas. We might draw this situation as at the right.
These sorts of molecules have a certain chemical architecture. Methane, for example, has a "tetrahedral" structure. See below.
So covalent bonding is a matter of sharing electrons. The number of electrons an atom has to share fits with which column it is in and to have a good molecule, you want a total of eight electrons shared.
7. There are of course many molecules that form covalent bonds that do not involve carbon. Nevertheless, there are so many that do involve carbon that a whole branch of chemistry, "organic chemistry," is dedicated to such molecules. Here are just a few of the types of molecules studied in organic chemistry:
- hydrocarbons - molecules involving carbon and hydrogen
- alcohols - have the subunit C-O-H
- ether - have an -O- in the middle
- aldehydes - have the subunit O=C-H (oxygen requires two bonds)
- ketones - have an O=C somewhere in the middle
- carboxylic acid - have a COOH
- esters - have the subunit O=C-O- in the middle
- alkene - have a C=C in the middle
 The SO4 ion is bound together mostly by the second type of bonding, mentioned below.
 In the very first Star Trek movie ever (1979), humans are referred to as "carbon based units."