After browsing Wikipedia, here's the Cliff's Notes:
The fatty acids we eat are all basically carboxylic acids (a C attached to an O with a double bond, and an OH with a single bond) with a carbon-chain attached to the spare bond.
If the chain is all single-bonds, with two hydrogens off of each carbon, then it's a saturated fat.
If the chain has one double bond, it's a monounsaturated fat. If it has more than one, it's polyunsaturated.
Most natural unsaturated fats are "cis-fats", meaning that at the two carbons with the double bond, the single hydrogens attached are both sticking off the chain on the same side of the molecule. This results in the molecule bending at that point, as the hydrogens repel, but are unable to cause the double-bonded carbons to twist.
In a partially-hydrogenated fat, a catalysis reaction has caused the section with the double bond to end up in a "trans" configuration, where the hydrogens are on opposite sides, allowing the molecule to straighten out. This straightened molecule will remain solid to much higher temperatures. It will also form horrible plaques in your arteries and give you a heart attack.
The "omega" terminology refers to where in the chain the first double-bond appears. If it's three carbons from the end of the tail, the fat is omega-3. It's not clear to me, from the articles, why our modern high-in-omega-6 diets are bad for us, though it's suggested that it may relate to the regulation of inflammation reactions (which play a role in immune system health, as well as all kinds of diseases involving improperly regulated inflammation, such as arthritis).
There. Now you can impress people at parties. Wikipedia rocks my small, self-centered universe.