Fats (triglyceride lipids) are made up of glycerol and fatty acids; these fatty acids are either saturated, monounsaturated or polyunsaturated. They have the same basic structure (a carbon backbone with hydrogen branches and a carboxyl end); the difference lies in the number of used or spare hydrogen bonds.
Fats are made from fatty acids (FAs); generally they are put into three major groups, but each group has many different types of that type with it. The main groups are saturated, monounsaturated and polyunsaturated fatty acids; here more about the FAs in each one.
There are four types of natural fat, saturated, mono-unsaturated and polyunsaturated and, less commonly, natural transfats; we also have unnatural transfats which are sadly far too common.