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A common fat is composed of glycerol and long chains of carboxylic acids called fatty acids. They are generally referred to as triglycerides. Triglycerides can be decomposed by adding NaOH, as shown in the little animation:
(the R group is usually a long chain of 16, 18, or more carbons)
Fatty acids are saturated if there are no double bonds in their chain, unsaturated if there is one double bond, and polyunsaturated if there is more than one double bond. Saturated fatty acids tend to be solid, since they form straight lines and can stack on top of each other. On the other hand, unsaturated fats are liquid at room temperature because since they have a double bond, their shape is bent and they can't form layers as well (this is why they're healthier).
Fatty acid anions can be the perfect emulsifiers because they have a polar "head" and a nonpolar "tail." In the fatty acid chain, the head is a COO- group. Since grease does not dissolve in water, soap works because the fatty acid anions form what are called micelles (those round sphere-like things, shown below) have their hydrophobic tails facing towards the center, and the grease becomes attracted to this:
Soap-water is just a suspension of micelles in water.
