Biodiesel can be made following various but similar processes. A simple process is to use vegetable oil, methanol and sodium hydroxide.
Again in general terms vegetable and animal fats and oils are triglycerides, containing glycerine. The biodiesel process turns the oils and fats into esters, separating out the glycerine. The glycerine sinks to the bottom and the biodiesel floats on top and can be syphoned off. The process is called transesterification, which substitutes alcohol for the glycerine in a chemical reaction, using a catalyst. The result is usually a 90% biodiesel and 10% glycerine.
source: www.biodieselpictures.com
1. Reacted biodiesel with glycerol on the bottom and biodiesel on top
2. Washed biodiesel with water on the bottom
3. Dried biodiesel

Why biodiesel doesnot add CO2 to environment ?

When burning petrodiesel the CO2 release in the atmosphere is in fact the CO2 that was “stored” underground in the petroleum layers for many years. Thus this process increases the overall CO2 in the atmosphere as the CO2 does not return underground.
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Using vegetable oils or animal fats as fuel for motor vehicles is in effect running them on solar energy. All biofuels, including ethanol, are derived from the conversion of sunlight to energy (carbohydrates) that takes place in the green leaves of plants.
Plants take up carbon dioxide (CO2) from the atmosphere; burning plant (or animal) products in an engine or burning it any other way releases the CO2 uptake back into the atmosphere, to be taken up again by other plants. The CO2 is recycled, atmospheric CO2 levels remain constant.
Thus biofuels do not increase global warming -- unlike fossil fuels, which release large amounts of new (or rather very old) CO2 which has been locked away from the atmosphere.
Biofuels and Water quality: Water pollution associated with petroleum product includes ground water contamination from spills. Biofuels can replace the most toxic parts of gasoline with fuels that quickly biodegrade in water, reducing the threat to waterways and groundwater. Spills or leaks of biofuels do not constitute an environmental hazard.

Diesel engine, Dr. Diesel.

“Dr. Diesel demonstrated his engine at the Exhibition Fair in Paris, France in 1898. This engine stood as a example of Diesel's vision because it was fueled by peanut oil - the "original" biodiesel. He thought that the utilization of a biomass based fuel was the real future of his engine. He hoped that it would provide a way for the smaller industries, farmers, and "common folk" a means of competing with the monopolizing industries, which controlled all energy production at that time, as well as serve as a n alternative for thin efficient fuel  consumption of the steam engine. The diesel engine was invented by German engineer  Rudolf Diesel (1858–1913). As a result of Diesel's vision, compression ignited engines were powered by a “bio” fuels or vegetable oil, until the 1920's and are being powered again, today, by biodiesel.”

What is biodiesel ?

Biodiesel is a cleaner burning alternative fuel produced from renewable vegetable oil resources such as soy beans, palm and waste vegetable oil (cooking oil) or any other source of organic oil (animal fat). Biodiesel is suitable for modern, high performance diesel engines. Biodiesel contains no petroleum product but can be blended with petroleum diesel to create a biodiesel blend. This type of fuel is gaining popularity not only due to its environmental advantages but also because of how easy it is to use as it can be used in current compression-ignition (diesel) engines with little or no modifications. Biodiesel is not only easy to use; it is biodegradable, non-toxic and free of sulfur and aromatics. Replacing conventional diesel with biodiesel in engines results in considerable reduction of unburned hydrocarbons, carbon monoxide, and particulate matter. Moreover, with flash point over 260°F (127 °C), biodiesel is safer to handle and to store than petroleum based diesel fuel, which has a flashpoint of around 125°F (52°C).
Biodiesel is defined as the mono-alkyl esters of fatty acids derived from vegetable oils or animal fats. In more general terms, biodiesel is the product you get when a vegetable oil or animal fat is chemically reacted with an alcohol to produce a new compound that is known as a fatty acid alkyl ester. A catalyst such as sodium or potassium hydroxide is required. Glycerol (glycerin) is produced as a byproduct. The process is known as transesterification.
Soybean oil and methanol are the most popular feedstock in the United States. Soybeans are a major U.S. crop and government subsidies/reduced taxes are available to make the fuel economically attractive to consumers who need or want to use a non-petroleum based fuel. In Europe, most biodiesel is made from rapeseed (Brassica napus) oil and methanol and it is known as rapeseed methyl esters (RME).

Advantages of Biodiesel

Biodiesel has some clear advantages over SVO: it works in any diesel, without any conversion or modifications to the engine or the fuel system -- just put it in and go. It also has better cold-weather properties than SVO (but not as good as petro-diesel). Unlike SVO, it's backed by many long-term tests in many countries, including millions of miles on the road. It has as well many advantages over petro-diesel.
• Biodiesel substantially reduces unburned hydrocarbons, carbon monoxide and particulate matter in exhaust fumes
• Sulphur dioxide emissions are eliminated (biodiesel contains no sulphur)
• Biodiesel is plant-based and adds no CO2 to the atmosphere. As a sustainable energy source it merely recycles carbon, with the help of the sun and photosynthesis.
• The ozone-forming potential of biodiesel emissions is nearly 50% less than conventional diesel fuel
• Nitrogen oxide (NOx) emissions could slightly increase but can be reduced to well below conventional diesel fuel levels by adjusting engine timing and other means
• Biodiesel can be used in any diesel engine
• Fuel economy is about the same as conventional diesel fuel
• Biodiesel has a high cetane rating, which improves engine performance: 20% biodiesel added to conventional diesel fuel improves the cetane rating 3 points, making it a Premium fuel
• Biodiesel can be mixed with ordinary petroleum diesel fuel in any proportion, with no need for a mixing additive.
• Even a small amount of biodiesel means cleaner emissions and better engine lubrication
• Biodiesel can be produced from any fat or vegetable oil, including waste cooking oil.

Diesel Engines

There are many types of internal combustion engines. Different types run on different type of fuel. Petrol, hydrogen, methanol, ethanol, liquefied petroleum gas (LPG), biodiesel as well as other fuels. But the most common are petrol with electronic ignition, spark ignition (SI) and the simpler diesel with compression ignition (CI). Yet both run on petroleum based products. Diesel from petroleum is also known as petrodiesel or even “dino-diesel”, when there is a need to distinguish it from diesel obtained from other sources such as biodiesel.
Petrol, also known as petroleum spirit (USA - gasoline, British- petrol) and diesel are both produced from fractional distillation of petroleum oil (fossil fuel). Diesel is a hydrocarbon mixture, obtained from crude oil between 200 °C and 350 °C at atmospheric pressure.
Both petrol and diesel can be wholly replaced by biofuels or mixed to certain extents. Petrol can be replaced by Ethanol and diesel can be replaced by biodiesel.