Cold Process is a method in which the maker mixes fixed oils with an alkali component (Sodium Hydroxide, aka Lye). When lye comes into contact with fatty acids (fats and oils) it creates an exothermic (it releases heat) reaction called saponification.
Saponification gets it's name from the Latin word "sapo", which means "soap". So saponification literally means "turning into soap". Every oil, fat and butter used has a "saponification value". This value is the amount of lye needed to complete the saponification process without having any lye left over in the bar (this is where the "lye soap isn't safe" argument goes out the window. The purpose of saponification is to neutralize and get rid of the lye's caustic properties. In short, making it safe to use. Chemistry is awesome). Before you ask, soap with lye left over is considered "Lye heavy" and not suitable for use on skin but this doesn't mean it has to be considered waste or dangerous. Lye heavy soap can be safely used as laundry soap.
Creating soap from scratch is a dangerous process. Lye is extremely caustic.
I emphasize the word caustic because it's important. Lye can cause serious damage if you don't show it the proper safety respect it deserves. This doesn't just mean protecting your skin and eyes, it means knowing what materials are safe to use during your soap making process.
After the saponification process is complete (over a period of a couple days give or take), you're left with a solid bar or loaf of soap. Once PH tested, you slice it how you like and let it sit for 4-6 weeks to finish curing. Sometimes longer depending on the type of soap you've made. Some oils take longer than others to fully cure. Castile soap (made with only olive oil) can take up to a year before it's fully cured! That's a long time to wait, so this process also requires patience.
Unlike store bought bar soap, I know what's going into my bars. I can take away or add an ingredient as I see fit to make that bar suitable for it's purpose.
I don't use palm oil in my cold process soaps! I do use murmuru butter which is a type of palm, but it's not the same palm that provides palm oil.
I believe honesty and transparency is the best policy. It's almost impossible to get all of your skin care ingredients "locally". We don't have African shea trees in America as far as I know. What I can do, however, is purchase my supplies from local suppliers. American based companies that are run by people just like you and me.
As much as we love lather, it's actually quite useless. I cut out additional bubbling agents. Some oils like palm and coconut help produce lather, but I don't use palm. Some of my soaps only produce a low amount of lather and I'm okay with that! Using low lather soap can be a weird thing to get used to, but the benefits far outweigh the strangeness!
Superfat is the percentage of fats left in the bar after the saponification process is complete. I make my bar soaps with 5% - 7% superfat. The higher the percentage, the gentler the soap is on skin. The downside to a high fat percentage is that it creates a softer bar of soap and should be used sooner rather than later. If you have really dry skin, you may opt to use a soap with a higher percentage of superfat. This is not to say that a low percentage will dry out your skin. That's not necessarily true at all. Toy around with it and find what works best for you!