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Second quantization refers to quantizing fields by expressing them as operator-valued distributions
The most elementary, or "semiclassical" treatments of quantum mechanics fix the number of particles and treat the field classically, including it as a parameter in the Hamiltonian or Lagrangian or whatever.
In the so-called "second-quantized" treatment, the treatment takes into account the quantum nature of the field in question, e.g. the electromagnetic field or, in solids, the phonon field. Particles are treated as field excitations (i.e. quanta). This gives a physically informative way to treat systems in which the number of particles of a given type change with respect to time, and a way to accurately treat quantum fields.
The term is objected to by many working physicists, and is considered deprecated. One is not quantizing "again", as the term "second" would suggest; one is merely shifting from a semiclassical treatment of a problem to a fully quantum-mechanical one.
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