Nitroglycerin is one of the earliest and simplest explosives used by man. It's also incredibly easy to make and very sensitive. It uses ingredients commonly found in drugstores and hardware stores. Mixed acid nitration is the process of using mixed acids to nitrate an organic material. Concentrated Nitric Acid and Sulfuric Acid are mixed together. Sulfuric acid loves to suck up any available water. So as your Nitric Acid nitrates the Glycerin the Sulfuric pulls in the Water by product. This also produces heat. Heat that is capable of causing the newly made Nitroglycerin to detonate. So this whole reaction needs to take place in a beaker that's surrounded by really cold ice either in a water bath or a acetone bath. But it can't be too cold because if it freezes the crystals as they break could also cause a detonation.
Synthesis instructions can be found here.. http://www.roguesci.org/chemlab/energetics/nitroglycerin.html
Some Trivia: Early Dynamite was made from Nitroglycerin not TNT. TNT and Nitroglycerin are totally different materials. Nitroglycerin was originally shipped in jars, like jam jars, which were put into wooden cases with sawdust to insulate them from the shock of riding along in traincars. One day a jar of nitroglycerine broke and spilled its contents into the sawdust. Nobel realized that the material might still be explosive and packed this Nitroglycerin coated sawdust into tubes inventing "Safety Dynamite".