What does it mean?
Triple milled, tallow based, veggie based, glycerin based and melt and pour. On a daily basis we see all these terms used on the B&B. Lets take a look at a few terms and processes and see if we can sort them out.
"Soap" has several definitions.
The technical definition of soap is the result of a chemical reaction between a fatty acid and an alkali solution. Fatty acids are the actual fat content of animal or vegetable oils; there are many different fatty acids which combine to make up different oils. The alkali solution is commonly known as lye and is usually sodium or potassium hydroxide. By this definition, soap must be made from animal or vegetable fats/oils and lye. However, note that in the finished product there is no free lye - it is all combined into the resulting soap. 
Hard Soap, Soft Soap?
So, the italic sentence in the above quote is the key to all that follows,
- Hard soap, is a soap that is made with sodium hydroxide.
- Soft soap, is a soap that made with potassium hydroxide, many of our old, pre reformulated shaving creams were of this variety.
Tallow based soaps are the result of the above process, using the fat of beeves as the fatty acids. tallow based soap can be hard soap or soft soap, milled or not milled.
Vegetable based soaps are the result of the above process, using the fat found in vegetable sources, palm, corn, soy, etc. Vegetable based soaps can be soft soap and hard soap. Milled or not milled.
Glycerin based soaps or melt and pour soaps can be more difficult to pin down, some are made from the by-products of the above process and are very fine quality,the true glycerin soaps.
Some are made from synthetic compounds like detergent. Many of the artisans soapers use a commercial soap base See here for some examples. The ingredients of these soaps bases run the gauntlet from-mostly real soap to mostly synthetic compounds and detergent.
Glycerin, triple milled?
As a general rule, glycerin based soaps are not milled.
Triple milled soap
Milling refers to a process of grinding, heating and cooling of the soap in a piece of equipment (the mill). The purpose of this process is to solidify and homogenize the soap, drive moisture out, and to add colorants,fragrance and any additional emollients. Triple milled soap would pass through this process 3 times.
After the soap is milled, it would be pressed into pucks or bars. The product is dense and very hard at this point.
Milling can be done with either vegetable based or tallow based soaps.
So whats this all mean for my shave?
Any of the above soaps or creams can be a terrific product, as has been discussed many times on the B&B, one type of product is not head and shoulders above the rest based on process alone. As a general rule, triple milled products tend to be of higher quality simply because the process to make them is more costly and the end product needs to have more value to the end user, and the vendor can charge accordingly.
Many non-milled soaps have a varying percentage of their weight as water, this is an inert ingredient in the soap. Milled soaps have almost all the water removed, so comparing ounce to ounce you can derive a greater value from milled soaps then from the non-milled even if the milled soaps are more costly initially.