There are loads of websites telling you how to make soap, hundreds of blog posts that start with massive preambles before getting down to brass tacks of how to do it. So lets do this a different way. I'm going to start with how I make soap, then I'm going to move on to why, and THEN I'm going to talk about whats actually happening.
Ok, tell me how to make soap.
For ingredients you're going to need oils (fats), lye (sodium hydroxide), water (rain water is ideal), and if you want to add some scent you'll be wanting a few drops of some essential oils (be sparing!), and perhaps dried herbs or spices. And if you really must colour it, use a crayon (take the paper off first, obviously).
And for equipment to work with, you need a huge stainless steel pan (I use my stock pot). I mean huge. No messing about with a little one, unless you want a disastrous boil over, potentially a kitchen fire and, at very least, a god awful mess. And you need some polypropylene jugs, and the most accurate scale you have - a digital scale working down to the gram is good. You need to stir the mixture as you make it - those rubber spatulas, the silicone ones, are ideal. But get a couple, they'll get hot as you work and swapping them is good. Lastly, you need something to pack the soap into while it sets - I use silicone molds.
Firstly, weigh everything out. My basic recipe is as follows:
400g Sunflower oil (ordinary cooking stuff)
400g Olive Oil (get the cheaper extra virgin stuff)
400g Coconut Oil (again, get the cheap stuff from a Chinese supermarket)
400ml rain water
180g sodium hydroxide
20 drops of essential oils, spices (optional)
Firstly, mix up your oils and get any other dry ingdredients ready. I often use a mix of nutmeg oil, cinnamon oil, a little pine oil, tea tree, and a hand full of coarsely ground allspice grains. Once these are ready, put them aside.
Now don your safety glasses and marigolds, and slowly pour the sodium hydroxide into the water, then stir it with one of your spatulas until it is dissolved. I've handled some of the nastiest chemicals you can imagine and some biological hazards you really don't even want to imagine, but most often its the every day chemicals that people have accidents with. Sodium hydroxide is one of them, it WILL blind you if you get it in your eyes. So don't. If it gets on your skin wash it off, and keep washing it.
Put the oil in the big pan, and start warming it up. Use your spatula to get the last of the oil out of what you weighed it in too. And when its warming, pour your sodium hydroxide solution into it and start stirring while you continue heating, being sure you get down to the bottom of the pan and keep the soap from sticking. Very soon, you get something like this...
It'll boil up and terrify you, but keep it on a heat and keep it stirring. If your spatula gets too hot, put it in a plastic jug to cool down and get your other one. Keep stirring, and soon it'll look like apple sauce. At that point, keep stirring...
Keep going. Heat it more gently if you like, but keep going. It'll start looking like mashed potato...
And now? Keep going, it'll start getting really thick and just at the bottom, where its hottest, it'll start to look a bit dry. Its done... 
Now get it off the heat and beat in the other ingredients. This is going to be hard work. But whether thats a crumbled up crayon, spices, oils, whatever it is, beat it until its combined and then start plopping it out into your mold. For this I use the two spatulas, and occasionally I risk pushing it down into the molds with the rubber gloves...
And thats it. Let it cool just enough so you can handle it, pop it out of the molds and put it on a plastic boards to cool. When its cool and hard, its ready to use.
You're kidding me, its that easy?
Yes. The whole process takes about, oh, I dunno, half an hour or so, maybe three quarters, until you can put the kettle on and wait for it to cool. But easy? You might find it hot, stressful, worrying, if you spill it you've got problems. But yes, its as simple and easy as that.
But why? You can buy soap...
Ok then don't. No skin off my nose. I make soap because its easy, quite fun, I can get just the soap I want with control over whats in it. Quite a rewarding thing to do. I've also always found all the kinds of soap you can buy just leave me itchy all over, this soap doesn't.
Whats actually happening then?
The chemistry of this is simple enough. You're taking a fat, chemically breaking it up, into fatty acids, and making the salt of that. The process is called saponification, and its quite fascinating when you read up about it.
Can I change the recipe or use other fats?
Yes, you can. You'll want to hunt down the saponification value of the oils you want to use though, otherwise you'll end up making something potentially very dangerous. The internet is full of sites that will give you those values, it won't take you long. But if you're going down that route, find some more in-depth reading first. This is a primer for doing your first batch, not a guide to all forms of soap making.
Is there a safer, lower stress way?
Yes. The process I've described above is hot processing. There is also a method called cold process with is easier and safer, but can take weeks of waiting. There's nothing wrong with doing it that way, but you do need to be able to put something aside somewhere safe and forget about it. Lots of advice about how to do that online, and its a common beginners way in. Maybe start here.
