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Fats are a type of lipid (which include oils and such) which are made from a glycerol head and a fatty acid tail. The fatty acid tail is mostly a chain of carbons attached to hydrogens. The structure of the fatty acids is divided into two classifications: saturated fats, and unsaturated fats.
Saturated fats have the carbon chain filled with hydrogens (each carbon is bonded to two other carbons and two hydrogens), so they form a straight line, and the molecules of the saturated fat can pack together well. Saturated fats tend to stay solid at high temperatures relative to unsaturated fats.
Unsaturated fats do not have each carbon filled with two hydrogens. Instead, the carbon will form a double bond with a neighboring carbon bond and bond with only one hydrogen, so there will be a kink in the fatty acid, making it angular (this is because the carbon's VSEPR structure would be trigonal planar). As a result, molecules of unsaturated fats do not pack together as well and stay liquid at higher temperatures relative to saturated fats.
Lipids do not dissolve in water (since they are nonpolar), so saturated fats can potentially clog arteries and blood vessels (more so than unsaturated fats, because they tend to be solid at higher temperatures).
EDIT: The health benefits are a bit more nuanced. There are also other reasons saturated fats are not healthy, compared to unsaturated fats. However, hydrogenating unsaturated fats may produce unsaturated fats with trans double bonds (transfat), which may contribute more to other health problems related to high fat consumption.