Hydrogen bonding is a special (and awesome) intermolecular force that can happen when hydrogen is attached to nitrogen, oxygen, or fluorine. All three of those are pretty small atoms and are also highly electronegative. They pull so much electron density away from the tiny hydrogen that the nucleus (a single proton) of the hydrogen is highly exposed and attracted to negatively charged things, letting it sort of stick to negatively charged bits of other molecules.
Fluorine is a halogen. I wrote about this, but it was so long ago that you probably forgot all about it. As a halogen, fluorine can only have a bond to one other thing. If that thing is hydrogen, then we have lots of hydrogen bonding fun, but there's only one compound for which this is possible, and that's HF (hydrogen fluoride), which isn't organic, so we won't be paying much attention to it right now.
Nitrogen and oxygen, unlike fluorine, can both be attached to hydrogen and have at least one bond to spare. So there are lots and lots of compounds that exhibit hydrogen bonding. Just look for hydrogen attached to nitrogen or oxygen. Hydrogen bonds are stronger than dipole-dipole forces and other intermolecular forces. In fact, they're partial covalent bonds, although still not nearly as strong as regular covalent bonds.
The classic picture for showing hydrogen bonds is with water, so I might as well just use one of those rather than trying to make my own picture. It's much easier and looks way better. Here's one...
Pretty much any property of water has something to do with hydrogen bonding. And it's important in lots of other compounds too. The structures of proteins and nucleic acids use plenty of hydrogen bonds. Hydrogen bonding is one of the reasons hydrogen is my favorite element. But just remember, it's stronger than other intermolecular forces.
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