The word itself says it all: 'natural' means that the ingredients are of natural origin. Within the term 'natural' there are many gradations, so you have products that are 100% natural to products that are only enriched with natural ingredients. On the contrary, there are the synthetic ingredients that are created through a chemical process.
Biological and organic mean the same thing. Such products use only natural ingredients produced without the use of synthetic pesticides. So biological or organic goes a step further than natural. Again, there are gradations such as the percentage of ingredients that are organic/organic. You can compare it to an unprocessed apple that is picked in the garden versus an apple that you buy in the supermarket.
The term 'cruelty-free' is also often mentioned on packaging. It means that the final product and the ingredients were not tested on animals. It is important to know that animal testing of cosmetics and ingredients is prohibited by law in Europe since 2013. Products with a cruelty-free claim therefore simply follow the legislation just like all other products in Europe.