Financial derivatives are contracts whose value depends on the value of other assets. For example, a contract might pay someone ten euros if the value of a share is above fifty dollars.
The market for derivatives became really large in the 1980s, so it was lucky that two economists, Black and Scholes, had worked out in the 1970s how to calculate their price if you know the price of the assets on which they are based. Their method is exact if the stock markets are perfect, that is, if they function without any delays, trading costs, or uncertain knowledge, and there are lots of assets, buyers, and sellers. The exactness of their formula gives the approach an advantage over competing theories.
Their central idea is simple - to build the derivative from the underlying assets and cash, so that you know its price because you know its components' prices. The process is called hedging in the literature, but synthesis, replication, or just copying would be a better description. There is technically competent algebra required as well to get exact formulae, but the idea of hedging, once grasped, is so direct that one feels like screaming with frustration at not inventing the formulae oneself. The world's best economists until 1973 could say the same.