Soap, in it's most basic and fundamental form, is the result of a chemical action between a fatty acid and a caustic alkali. (See How Is Soap Made?) However, the method by which that chemical action is accomplished can vary widely. There are some similarities and vast differences between commercial and handmade soaps.
Generally, when one speaks of commercial soap one is referring to soap made in huge batches (100,000 pounds or more) in a mechanized process where the glycerin is removed and fillers and sometimes synthetic detergents added. Often a commercial soap manufacturing plant has facilities for refining oils, recovering and purifying glycerin (which is then sold separately) and producing soap in various final forms.
Many companies that produce soap and soap related products (such as shampoos, cleaning agents and even toothpaste) buy the basic soap and then add their own ingredients and packaging to the final product.
Handmade soap, on the other hand, is generally thought of as soap made in smaller batches with personal attention, where the naturally occurring glycerin is retained in the soap. Fillers or detergents are rarely added to hand-crafted soaps because these require facilities and chemistry unavailable to (and unwanted by) most hand-crafted soap makers.
Often the hand-crafted soap maker does the entire process from start to finish, including manufacture, scenting, curing, cutting, trimming and packaging.
There is also a hybrid form of soapmaking, called melt and pour. It is a commercially produced soap base which is chemically formulated (through the use of various additives) to be able to be melted and then poured without changing its consistency. The soap base is then purchased by a soap maker who melts it down, adds colors, scents and other artistic touches and packages it for final sale. Melt and Pour soaps can be true soaps and are generally considered to be "handmade" if more than 50% of the entire process from recipe formulation through to packaging is done by hand.