There is probably an infinite number of possible environmental effects one can describe with the Lindblad equation. But one can gain some understanding of the collapse operators $A$ by considering some simple cases.
If $A$ is an orthogonal projecton-operator on a subspace $H_+$ then the action of the Lindblad equation will be to kill the coherences (off-diagonal terms in the density matrix $\rho$) between the states in $H_+$ and $H_-$ (where the Hilbert space is $H = H_+ \oplus H_-$).
If $A$ is a rotation-operator (I have no better name for it) of the form $|\psi\rangle\langle\chi|$ then the action of the Lindblad equation will be to kill all the probabilities related to states including $|\chi\rangle$ and move them to corresponding states including $|\psi\rangle$.
Since one can generate a great deal of different operators $A$ by combining the described rotations and projections this can help to interpret various environmental effects in terms of the collapse operators.
The constant $\gamma$ (which must stand in front of the brackets!) describes only the strength of the collapse process. Essentially, it determines the time-scale of the collapse process.