When a quantity can only take on integer multiples of some base value, those units are called quanta (sg. quantum), from the Latin for quantity. An important discovery of modern physics is that most if not all fields are quantized, hence the field of quantum physics. Quantization often arises from boundary value problems.
In physics, quantization refers to the formulation of a classical theory in the formalism of quantum physics. Even though classical physics stems from quantum theory, the build up of a quantum theory is often made the other way around, starting from existing classical physics to derive the more fundamental quantum counterpart. For instance one can speak of the quantization of the electromagnetic field.
2nd quantization refers to a special formalism of quantum theory suited to deal with variable number of particles. It pertains to quantum field theory and draws its name from a loose understanding of the formalism as quantifying once more an already quantized theory.