Consider how the states available to the electrons are actually populated. According to classical statistics at absolute zero, T = 0K, all the particles would occupy the lowest state, where kinetic energy is zero. However, electrons do not obey classical statistics. Instead, they obey a quantum statistics called Fermi-Dirac statistics. Particles like electrons that obey the Fermi-Dirac statistics are known as fermions. (they are indistinguishable particles and obey Pauli’s Exclusion Principle)
Fermi-Dirac statistics takes into account the exclusion principle. According to the principle, no two electrons in the metal can have the same set of quantum numbers. Hence, in each of the energy states, there can be at most 2 electrons (one with spin up and one with spin down).
At T = 0K, all the possible energy levels would be filled up with two electrons each up to a maximum level called the Fermi Level. The energy of the state at the Fermi level is known as the Fermi energy, EF.