A chemical bond is what holds molecules together. When you have something like H2O, a chemical bond holds the two hydrogen atoms to the oxygen atom. By this definition a hydrogen bond isn't strictly a chemical bond, as it does not hold molecules together, but rather is a way to describe the interaction between a large group of molecules. However, they are related, as a sort of "Bonding" occurs between multiple molecules. Behold, the molecular shape of water!
You'll notice that, with respect to the atoms involved, it has what is called a "Bent" shape that resembles the shape of the letter "V". The four dots around the oxygen atom represent electrons that the oxygen carries around with it. The important thing to know about those electrons in this case is that they are negatively charged, like magnets, which have both a positive and a negative side.
Or a North and South side, as in this picture. Same idea. In fact, if you've played with magnets, water behaves in much the same way: The negative side of water is attracted to the positive side of water. The negative side of water is the side with the oxygen, because oxygen "likes" to carry around electrons (relative to hydrogen). The positive side of water is around the two hydrogen atoms for two reasons: Hydrogen atoms are single protons, which have a positive charge, and as stated before, the oxygen atom "likes" to carry around negative charge much more than hydrogen does. So the oxygen atom not only has the four electrons that it normally carries around, but it will also carry both hydrogens' electrons around. This causes the entire water molecule to become "polar" in the same way that the bar magnets above are polar: With a North and a South side.
Hydrogen bonding is this sort of interaction: Where one side of a molecule will have hydrogen atoms attached to atoms, like Oxygen, which will carry much more negative charge than hydrogen will. This causes a polarity on the molecule, and then large groups of that molecule will interact with itself, where the negative side will be attracted to the positive side. This won't cause true chemical bonds, as they aren't new molecules, but the interaction is enough to have an effect on macroscopic observations, such as boiling point.
To relate this back to the post on distillation: you'll notice from the diagrams in the previous post that drinking alcohol also happens to have an oxygen atom with a hydrogen atom attached to it, which makes it suspiciously similar to alcohol. In fact, this is the case, and some hydrogen bonding occurs in drinking alcohol. However, you'll also notice that there are two hydrogen atoms attached to the oxygen in water, and only one in the case of drinking alcohol. This allows for multiple hydrogen bonds to form, which makes water more attracted to itself than alcohol is to itself. Because water is attracted to itself than alcohol is to itself, it takes more energy (and hence a greater temperature) to cause it to boil. So in the distillation process, alcohol will evaporate before water because of the effects of hydrogen bonding are greater on boiling point than the molecular weights.
This sort of explanation is the essence of chemistry. There are a number of physical things one can measure. There are a number of attributes to a given compound. But the desired end goal is to find a molecular explanation for a macroscopic observation -- something that the hydrogen bond easily does in this case. Also, for further reading, check out the effects of the hydrogen bond on DNA configuration and the density of ice. It has a great deal of explanatory power across several differing areas of study, as well as theoretical justification in physics. These are all the makings of great scientific facts.
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