As title says, how does a system interact with environment? I realize that this interaction can lead to interference terms and non-diagonal terms in density matrix being reduced (quantum decoherence). But what exactly is this interaction of system and environment? What would be the example of this interaction and when does this interaction occur?
The interaction between the system and environment is really just any plain old interaction you like. For instance, if you can have electron-EM coupling, or spin-spin interaction or whatever. What is key to decoherence is that once you write the Hamiltonian for the total system+environment you trace over the environment degrees of freedom - this corresponds to ignorance about the microscopic properties of the environment. @yuggib said that the Lindblad master equation usually describes open quantum systems, which is not quite true. Lindblad is very special and nice (it is defines a completely positive trace preserving map and is Markovian), but decoherence (and thermalization!) is quite a generic process and does not occur only under very special circumstances.
EDIT: Now I see from your comments that you're confused about locality of interaction. The interaction between the system and the environment can be perfectly local, e.g., you have a spin chain coupled only at one site to the environment. The system need not interact with the all (or even many) degrees of freedom of the whole environment for decoherence What you need is the environment to be very large compared to the system.