TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Method: The fatty acid raw material is fatty acid distillate, which is produced as a by-product during the process of refining crude vegetable oil to obtain refined vegetable oil. The fatty acid distillate contains fatty acid (65-95 wt.%), which has carbon atom number of aliphatic part of 14-24. The method further involves firstly refining a crude fatty acid alkyl ester produced at the esterifying step of fatty acid with alcohol, by feeding the crude fatty acid alkyl ester to a distillation column and then distilling impurities having low boiling point from the crude fatty acid alkyl ester, by maintaining the temperature of the lower part of the distillation column to 150-250degreesC at the vacuum condition of 0.1-150 torr; and secondly refining the firstly refined fatty acid alkyl ester by feeding the firstly refined fatty acid alkyl ester to a distillation column, by maintaining the temperature of the lower part of the distillation column to be 200-300degreesC at the vacuum condition of 0.1-150 torr for obtaining fatty acid which has 14-24 of carbon atom number of aliphatic part by distillation and removing residues. The alcohol is introduced by an amount of 0.5-5 times by weight with respect to the introduced amount of the fatty acid raw material, the reaction temperature for the esterifying step is 250-320degreesC, and the reaction pressure for the esterifying step is atmospheric pressure to 5 bar. The excess alcohol which is unreacted at the esterifying step is extracted through the upper part of a reactor with water produced during the esterifying step, and is introduced into an alcohol recovery section installed outside of the reactor, and is then recycled to the reactor by distillation. Preferred Apparatus: The second reactor comprises: a reactor body; and at least one comparting wall which divides the interior of the reactor body into at least two compartments, and the upper part of the comparting wall is open so that reactants overflow over the comparting wall. The divided compartments in the second reactor are arranged in the form of vertical stack, and a liquid guide path and a gas guide path are installed on the comparting wall. The compartments includes a first compartment and a second compartment which is located under the first compartment. The liquid guide path and the gas guide path penetrate the comparting wall, one end of the liquid guide path is located at the height corresponding to the surface of reactants located in the first compartment, and the other end of the liquid guide path is located at the height which is higher than the surface of reactants located in the second compartment, one end of the gas guide path is located at the height which is higher than the surface of reactant located in the first compartment, and the other end of the gas guide path is located at the height which is higher than the surface of reactant located in the second compartment. The apparatus comprises a reacting section having at least one reactor, for reacting fatty acid raw material with alcohol to produce crude fatty acid alkyl ester; a first refining section for removing impurities having low boiling point from the alkyl ester by distillation to produce a firstly refined fatty acid alkyl ester; and a second refining section for the distillation of the firstly refined fatty acid alkyl ester to obtain fatty acid alkyl ester and removing residues, where an alcohol recovery section is connected to the top of the reacting section, for recycling an excess alcohol and removing water produced at the reacting section. The reactor is a continuous stirred tank reactor on which a stirrer is installed, or a plug flow reactor in which a baffle is mounted. EXAMPLE - Fatty acid methyl ester (continuous process) was prepared as follows: The esterification reaction was carried out by two-step reaction using reactors. A first reactor was controlled to temperature of 300degreesC and pressure of 3 bar. Fatty acid distillate (1 kg) obtained from distilling crude palm oil, and methanol (1 kg) were introduced into the reactor and the reaction was carried out for 2 hours. Then, the temperature of the second reactor was maintained to be same with that of the first reactor, and the pressure was reduced to atmospheric pressure, and the reaction in the second reactor was carried out for 2 hours. For the reaction in the second reactor, methanol (2 kg) was divided by 3, and each divided part was introduced into each compartment. The reaction product was transferred to the first refining section. In the first refining section, the lower part of the distillation column was maintained at 200degreesC and 20 torr, and the impurities having low boiling point of 3 wt.% of the introduced feed was removed by distillation through the upper part of the distillation column. The firstly refined product was transferred to the second refining section. In the second refining section, the lower part of the distillation column was maintained at 250degreesC and 20 torr, and fatty acid methyl ester of 80 wt.% of the introduced feed was obtained by distillation. On the other hand, the water produced in the reaction section and the unreacted methanol was transferred to the alcohol recovery section, and methanol was recovered by distillation, and the recovered methanol was recycled and reused. The conversion ratio for fatty acid methyl ester was 99.7%.