What are the enegy units and what are the maas units in E=mc^2?
strictly speaking, the units are not defined, a priori. The equation remains true without regard to the units of measurement, as long as the units of measurement are internally consistent. The standard units are from the kg-m-s (kilogram-meter-second), or SI, system (sometimes identified as MKS, which contrasts with CGS=cm-gram-second). They do not have to be. You can muddle through using feet and pounds if you choose.
In SI, E is usually expressed as joules. 1 joule=1 newton-meter=1 kg*m^2/sec^2; the mc^2 side=kg*(m/sec)^2; same thing. All that is important is that all of the units correspond. you cannot have an equivalence if you measure different components of an equation according to incompatible measurement systems. You would have to convert some of the measurements.
The usual: Joules and kilograms typically. But for the small stuff (e.g., particles) we often use electron volts (eV) for the energy and, get this, m ~ eV/c^2 electron volts over c^2 for the mass.
By Newton's laws, Force is the product of mass and acceleration and force is measured in Newton eg. (5kg * 9.81 m/s-2) Newton.
Energy is confined resonant linear oscillatory forces and is measured in Newton-meter.
Here energy is confined forces so distance of one meter is involved.
One newton meter is equal to One Joule.
Mass is in kilo grams.
Energy is in Joules.