Energy — as a mathematical concept rather than a vague intuition — was not in Newton's vocabulary. The Principia uses words like vis viva ("living force," Leibniz's term, roughly mv²) and Newton's own quantity of motion (mv, what we now call momentum). The modern unification — that energy in all its forms (kinetic, potential, thermal, chemical, electromagnetic, nuclear) is one quantity that can change between forms but is never created or destroyed — took two centuries of false starts to assemble. The first law of thermodynamics, formalized in the 1840s by Mayer, Joule, Helmholtz, and others, finally pinned it down. Energy conservation is now considered the most fundamental principle in physics — more fundamental than Newton's laws themselves, because it survived every twentieth-century revolution that overthrew the rest of classical mechanics.
Energy is a scalar quantity associated with the state of a physical system. Kinetic energy (½mv²) is the energy of motion. Potential energy depends on position in a field of force — gravitational potential (mgh near Earth's surface, more generally −GMm/r), elastic potential (½kx² for a spring), electrostatic potential. Conservation of energy: in an isolated system, the total energy is constant; what changes is how much sits in each form. A pendulum swings between potential energy at the high points and kinetic energy at the low points, conserving the total. A roller coaster trades altitude for speed and back. The work-energy theorem connects energy to Newton's laws: the work done by a force on an object equals the change in its kinetic energy, and work = ∫ F · dx — the integral of force along the path. Different physical regimes have additional energy types: thermal energy (the kinetic energy of microscopic random motion, related to temperature), chemical energy (stored in molecular bonds), electromagnetic energy (in fields), rest-mass energy (E = mc², Einstein 1905), nuclear binding energy (mass-energy released in fusion or fission). Noether's theorem (1918) gives a deep reason for the conservation: energy is conserved because the laws of physics are invariant under time translation — they don't change from one moment to the next. Friction, which appears to violate energy conservation by stopping motion, in fact only converts kinetic energy into thermal energy — the molecules of the surface and of the object itself heat up by exactly the amount of kinetic energy lost. Once you measure the heat, the books balance again. The realization that heat is energy — Joule's experiments in the 1840s with paddle-wheels stirring water — was the conceptual move that finally made the first law of thermodynamics a precise statement.
Energy conservation is invoked daily in nearly every applied physical science. Power-grid management tracks energy flows in real time. Climate modeling tracks energy flows between the sun, atmosphere, oceans, and outer space. Battery technology is engineering of chemical-to-electrical energy conversion at high efficiency. Renewable-energy economics turns on conversion efficiencies (a solar panel is 20–25% sunlight-to-electricity; a wind turbine is 35–45% wind-to-electricity). Nuclear reactor design is energy management at the gigawatt scale. E = mc² — energy and mass are the same thing in different units — is the principle behind every nuclear technology. The principle of energy conservation has survived every revolution in physics — quantum mechanics modified what counts as a measurable quantity, special relativity unified mass with energy, but the total still doesn't change.