What is natural width in particle physics?

I think they refer to the decay width of the Higgs boson. By "natural", I think, they mean the ideal width that is predicted by the theory and that would be observed in the plots if everything was measured with an infinite precision. In reality the detectors are not perfect and measure the energy/momentum of particles with some error, which leades to smearing of the peak.

To add onto what has already been said: the "theory" width tells you how far away the mass of a particle can be from its nominal mass. That is an energy spread. And that spread is related to the lifetime. Due to the uncertainty principle, a short lifetime requires a large energy range (or mass range or natural/theory width, whichever term you prefer). And vice versa.