Without going into the precise chemistry, soap is made from combining an acid (oil/s) and an alkali (sodium hydroxide), known as lye, in precise proportions. It is easy to see which oils and fats different soaps are made from. For instance, if you make coconut oil soap, the name for the soap which will appear on the ingredients list will be 'sodium cocoate', palm oil is 'sodium palmate' and so-on. If you look at the ingredients listings of soaps you can fairly easily see which oils they are made from.
We choose only to use pure vegetable oils in our soaps, although animal fats can be used. One popular animal fat used in many soaps is Beef Tallow, seen on the label as 'sodium tallowate', so any vegetarians amongst you will now be able to spot whether you are happy using your current soap or not.