How is handmade soap made? Let's talk about the basics!
Soap is made using three main ingredients: Fat, liquid, and lye. There are variations of each of these items and they all produce a different end product. Basically though, the fat is melted, the liquid and lye are combined, and then they are added into the melted fat. Once it is mixed together, the resulting soap will be poured into a mold of some sort and left to sit for several days to set up, cut, and then left again to sit for 6 weeks to cure.
That's the short version! Let's add in a few more details. This is a fun process, but it can be dangerous, and it does take some science to produce a usable bar of soap.
FAT-
The fat you choose to use for your soap depends on what you're looking for in the end product. Coconut oil creates a hard bar that works great for dishes, laundry, and household cleaning. A 100% coconut oil bar would dry your skin and hair out so you wouldn't want to use these for shower or hand soap. Olive oil, on the other hand, is very moisturizing. 100% olive oil soap is known as Castile soap, most likely named for the area that it purportedly originated in Spain. Castile soap is an excellent all-purpose soap- probably my favorite, actually! It doesn't have a rich lather, so for those who are used to the "suds equals clean" mentality, castile soap may take some getting used to. Animal fats, such as lard and tallow, are also popular soap-making fats. They possess qualities that only animal fats can like vitamin D, and they create a nice creamy lather. Tallow and lard are not quite as easily attainable as their plant-based counterparts, which make them somewhat less desirable as a main soap ingredient.
While these are common soap-making fats, there are plenty of others as well! Sunflower, almond, jojoba, shea and cocoa butters, and palm oil are found often in soap recipes too.
Liquid- 
Just as there are lots of options for fats, there are quite a few options for the liquid portion of your soap as well. Distilled water is probably the most popular for its ease of use, and as it is free of contaminants, it will create a really pure bar soap, or clear liquid soap. Milk is another popular ingredient- and for good reason! Milk is high in fat which contributes to the lather of the bar and the moisturizing affect for your skin. Soap bars made with milk (and especially goat's milk) are praised for their help in clearing up chronic dry skin conditions. Coffee, aloe vera juice, and herbal teas are fun options for soap-making too.
Lye-
Lastly is the lye. There are 2 types of lye: sodium hydroxide and potassium hydroxide. Sodium hydroxide produces a solid, hard bar of soap, whereas potassium hydroxide produces liquid soap. Lye is a natural by-product made from soaking wood ash in water for a time. This is a whole science in and of itself! The chemical reaction that the lye facilitates is called "saponification". This is what turns all of those ingredients into a bar of soap.
The Merriam-Webster defintion for saponification is: "the hydrolysis of a fat by an alkali with the formation of a soap and glycerol
The variety of ingredients and add-ins for your soap makes for an extremely fun process that you can customize to fit your preferences and needs. There are no worries about allergens, unnatural chemicals as ingredients, or scents you may not like. You can make it to be whatever you want! This article barely even scratches the surface of the entire world that is soap-making, and I look forward to sharing more!