One of the fundamental laws of he universe is that energy is neither created nor destroyed -- it only changes forms. Consequently, many formulas for energy exist. In order to understand how these formulas are expressions of the same thing, it's important to first understand what physicists mean when they talk about energy. It's a notion that has its roots in the concepts of classical physics as elucidated by Sir Isaac Newton.
Force and Work
Newton's three laws of motion form the basis for classical physics. The first law defines force as that which causes motion, and the second law relates the force acting on an object to the acceleration it undergoes. If a force (F) accelerates a body through a distance (d), it does an amount of work (W) equal to the force multiplied by the distance times a factor that accounts for the angle between them (theta). As a mathematical expression, this means W = F•d•(the cosine of theta). The metric units for force are newtons, those for distance are meters and those for work are newton-meters, or joules. Energy is the capacity for doing work, and it is also expressed in joules.
Kinetic and Potential Energy
An object in motion possesses its energy of movement, which is equivalent to the work that would be required to bring it to rest. This is called its kinetic energy, and it is dependent on the square of the object's velocity (v) as well as one half of its mass (m). Mathematically, this is expressed as E(k) = 1/2•m•v•v. An object at rest in Earth's gravitational field possesses potential energy by virtue of its altitude; if it were to fall freely, it would gain kinetic energy equal to this potential energy. Potential energy is dependent on the object's mass, its height (h) and the acceleration due to gravity (g). Mathematically, this is E(p) = m•h•g.
The calculation of energy in electrical systems depends on the amount of current flowing through a conductor (I) in amperes, as well as on the electrical potential, or voltage (V), driving the current, in volts. Multiplying these two parameters gives the power of the electricity (P) in watts, and multiplying P by the time during which the electricity flows (t) in seconds gives the amount of electrical energy in the system, in joules. The mathematical expression for electrical energy in a conducting circuit is E(e) = P•t = V•I•t. According to this relationship, leaving a 100-watt lightbulb burning for one minute expends 6,000 joules of energy. This is equivalent to the amount of kinetic energy a 1-kilogram rock would have if you dropped it from a height of 612 meters (ignoring air friction).
Some Other Forms of Energy
The light we see is an electromagnetic phenomenon that has energy by virtue of the vibrations of packets of waves called photons. German physicist Max Planck determined that the energy of a photon is proportional the frequency (f) with which it vibrates, and he calculated the constant of proportionality (h), which is called Plank's Constant in his honor. The expression for the energy of a photon is thus E(p) = h•f. According to Albert Einstein's Theory of Relativity, each particle of matter has inherent potential energy proportional to the particle's mass and the square of the speed of light (c). The relevant expression is E(m) = m•c^2. Einstein's calculations were corroborated by the development of the atomic bomb.