You'll be disappointed to know that the official name for the law of conservation of energy is simply "The First Law of Thermodynamics" and yet, it is among the most important laws of physics. It is a relatively new law; it was discovered less than 200 years ago and it is a remarkable law in the sense that it involves every branch of physics. In some sense it is a unifying principle that can be applied to various kinds of phenomena.

Historically, the concept of energy was born with the branch of mechanics within physics with no apparent relation to other branches such as electromagnetism, optics or, say, acoustics (sound). Then it was found that energy is also involved in every branch of physics. As it turned out, it was later found to involve chemical phenomena as well. Today we know that even biology has something to do with energy. Energy comes in various forms such as mechanical, electromagnetic (including light), nuclear, chemical and thermal. This law states that no matter what forms of energy you begin with and what forms of energy you end up with in ANY process (including chemical and biological processes), the total amount of energy you have at the end is exactly the same amount you had at the beginning.

As to the changes on one system induced by another system, in general, whenever there is a difference in the value of certain kind of physical quantities (such as pressure, temperature or electric potential for example) between the two systems, if you let them interact with each other by placing them in contact (such as mechanical, thermal or electrical contact, depending on what is different in the two systems) then, both systems will change until that difference in value vanishes. For example, if the two systems in question differ in temperature and if you then put them in thermal contact, then the hot one will cool down and the cold one will heat up until both reach the same temperature. In the process, energy (in heat form) will be transferred from the hot one to the cold one.

This is usually called "Conservation of energy" or "The law of conservation of energy."It turns out that it's not quite correct, because energy can be converted into matter and vice versa. (That's the origin of Einstein's famous formula, E=m*c².The "E" is the amount of energy, "m" is the mass of matter, and "c" is the speed of light, which is a very large number. So you get a *very* large amount of energy for small amount of matter.) The modern law is usually called something like "Conservation of matter and energy."