What is the nature of the state of a 2 particle system? If I want do describe two particles #1 and #2 within Quantum Mechancis - let them be different in some way (Spin for example) - how do "I" decide if the total state of the system should be an entangled state, a product state or some density matrix?
I am concerned about this because I read that for some reason the universe is technically in an entangled state with everything and if we ignore it we should write every state as a density matrix (because we end up with a statistical mixture, if we ignore the universe). But this would mean that writing states as Dirac Brakets would be always wrong. 
Edit:
I can build up a two particle system ether as product-state or as an entangled state.
Say I want to mix an electron with the rest of the universe.
If I choose a product state I can ignore the universe and just look at the electron without losing any information. My electron description is complete without the universe.
If I choose an entangled state for the electron and the universe and I again want to focus only on the electron I end up with a statistical mixture. That I can ONLY write as a density matrix because I choose to ignore the universe I "forget" some information about the state. I add to the quantum probability the probability due to my incomplete knowledge of the actual electron state. This is not possible with Diracs Braket.
My question is how do I choose with what kind of state I start? 
 A: Density matrices and linear combinations of brakets are two different but equivalent ways of representing quantum states. On the other hand, entangled states and product states (if we share the same understanding of the terminology) are two different kinds of quantum states. So you're mixing up two questions: first whether a given state is entangled or not, and second, whether it should be represented as a density matrix or as Dirac brakets. If you keep them separate, I think that resolves your issue.
A: Extending David Z's answer: 
If you want to answer your question, you can ask the problem differently: your particle is independent or it is part of the system, ergo it is interacting with the system.
In entangled state, although you loose concrete information about your particle, but you get something much more greater: relational information.
I suppose you understand what an entangled state is, so I won't go into details, but if you want I will explain it with pleasure.
So if your particles are interacting with each other, and one's action has a direct effect on the other, then you start as an entangled state, otherwise as a product state. If you have a complex system then entanglement is often broken and remade ( like atom ionization ).
