Its been awhile since I've taken Organic Chemistry and Biochemistry, but this is what I remember.
Fats are organic molecules that contain carbon chains. The carbon atoms can be attached to each other either with a single or double bond. They would look something like this.
Single Bonded carbon atoms C-C-C-C
Double bonded carbon atoms C=C
When all the carbons in a fat molecule are bonded to each other with single bonds, we say its a saturated fat. Saturated fats are considered bad, because the way they are structured in 3 dimensional space. They are pretty straight, and pack well together, so they tend to clog arteries.
When there are double bonds present, the fat is considered unsaturated. If there is only one double bond its a mono-unasaturated fat. The double bond usually causes the molecule to be bent at an angle, which makes it difficult for the fat to pack together. Thus, less likely to clog arteries.
If there is more then one double bond, its called a poly-unsaturated fat. These molecules are also usually bent due to the double bonds, and they are also less likely to pack together and clog arteries.
Which is better mono or poly? Well, my professor never really said. I've heard its good to eat both.
Finally, there are trans fatty acids. These are fats that have a double bond, but are not bent at an angle. This can happen when the carbons next to the double bonded carbon pair are on opposite sides. Like saturated fats, these tend to clog arteries.