First, there is cold process soap. This is the kind I make. Cold process soap is made by combining an acid, plant oil, with a base, lye. When the lye solution is made, the chemical reaction produces a lot of heat. If the plant oils that are used are solid - such as coconut oil - heat is used to melt them. This type of soap is called cold process because no additional heat is added to the reaction. The process of saponification relies on the heat from the lye.
Very similar to cold process is hot process soap. The only difference here is that extra heat is added. This is usually in the form of a crockpot, but it can be done in a double boiler or in the oven. The soap "cooks" after the oils and lye solution are combined. It ends up getting thick like mashed potatoes, which makes it more difficult to mold, but gives the finished soap a rustic look. Also, the soap is technically ready to use after it is cooled. The lye reaction is complete due to the added heat. Many soapers who use hot process will still allow their soap to dry for several weeks.
Either cold process or hot process soap can be made into hand milled soap. Soapers call this rebatching. In this process, soap is grated or cut into small pieces and melted. The texture of the melted soap is very thick, so the appearance of the finished soap is very similar to hot process. Many times rebatching is used to salvage a goof up or to protect a very delicate fragrance or color from the effects of lye. You can find grated soap to purchase if you are interested in making soap, but don't want to use lye directly.
There is also melt and pour soap, or pre-made soap base. This is similar to rebatching because it is melted before color and fragrance are added. However, melt and pour base is made using different ingredients and melts smoothly. It is easy to pour and comes in many varieties, including clear. There are melt and pour bases that are made from natural ingredients, but the composition is different than cold or hot process soap. Many soap bases use detergents and chemical foaming agents and are not true soap. This is another way soapers can create soap without using lye themselves.
Liquid soap can be handmade as well. The process for making liquid soap is much like hot process. Oils are combined with lye - although potassium hydroxide is used instead of sodium hydroxide. Then, the soap batter is cooked before colors and fragrances are added and it is diluted to the right consistency.
Finally, there are some unique modifications to cold process, hot process, and liquid soap. Whipped soap is made using the cold process method at lower temperatures. The oils are whipped before the cooled lye solution is added. It results in a light, airy bar of soap that will float. Cream soap is somewhere between a solid and liquid soap. It is made using a combination of the two different types of lye and cooked similar to hot process or liquid soap. Many people like cream soaps for shaving. Soap jellies are made from combining liquid soap with gelatin. They work just like a regular bar of soap, but have a unique giggly texture.
I have only made cold process soap. I've also rebatched the soap that I made. Lately I have been doing some research into other types of soap and I'm looking forward to experimenting a little bit next year. My first exploration will probably be whipped soap since that's the closest to what I do now. Be sure to check out the shop - here - every now and then as I add new products as often as I can.