Making biodiesel from vegetable oil is fairly straight forward as it is the same procedure for performing the GC analysis on fats to quantify the various fatty acids (in the form of methyl esters) that were once attached to a glycerol molecule as the backbone, but just on a much larger scale.
Fat is commonly in the form of triacylglycerols (commonly called triglycerides) so three fatty acids linked by ester bond to a common glycerol. Triglycerides are too thick, viscous with low volatility to serve as a fuel but by pulling the individual fatty acids off of the glycerol backbone, they are now in a similar range as diesel (slightly higher with 16 and 18 carbons vs diesels 12-15 carbon). As free fatty acids they are not good as they have a charged group which can bind water and muck up the engine. So you cap the free fatty acid with a ester making it far more oil soluble and increase its volatility. This is the transesterification process.
So by approximately adding 20% methanol (to make methyl esters or ethaol to make ethyl esters) to 80% vegetable oil with a very exact amount of sodium hydroxide in methanol determined based on titrating a small amount of the vegetable oil to endpoint, and then heating to 50-60 C and recirculating through a tank (people often use modified water heaters that have a spout and valve at bottom for draining) you will pull the fatty acids off the larger triglyceride molecules and the glycerol will sediment to the bottom overnight. You can now drain off the approximately 10% glycerol created leaving 90% lower MW fatty acid methyl esters created (FAMEs or fatty acid ethyl esters if ethanol was used). This is your biodiesel! You should notice it is a lot thinner/less viscous than the original vegetable oil, and this makes all the difference in an engine system to maximize combustion. The glycerol can be discarded, or if distilled to remove the excess methanol can be used in making hand creams and moisturizers.