In theoretical physics, BPS states are massive representations of an extended supersymmetry algebra with mass equal to the supersymmetry central charge Z. Quantum mechanically, if the supersymmetry does not get broken, the mass is exactly equal to the modulus of Z. Their importance arises as the multiplets are shorter than for generic massive representations, the states are stable and the mass formulae is exact.
See also Bogomol'nyi–Prasad–Sommerfield bound and short supermultiplet.