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A state that has been perturbed, ie, disturbed slightly. Often, the trick is that if you have a state whose time evolution is required, and a very similar state's time evolution is known, then we can try to model it such that the former is a perturbed state of latter and go ahead and solve it, that is , a small disturbance like electric or magnetic field is imparted to the latter state, which makes it the former state.
Let me be more explicit, a Hamiltonian H can be considered to be made of 2 parts, H = H0 + delta H , where H0 is a known Hamiltonian, and delta H is a disturbance imparted to it, to make it H. And as I said earlier, typically the disturbance is either an electric field or a magnetic field.
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