Here's your typical experiment. Suppose these particles are electrons/positrons and they are sent one pair at a time. At O, the two particles are released from rest and sent in opposite directions, and of course because of the conservation of angular momentum, they will have opposite spins as they move toward A and B, where observers are stationed.
Let's say those particles are going to be passed through a magnetic field aligned in the direction of the Z-axis, giving each either a spin up (+) or a spin down (-). Alice will record each individual electron as they arrived, along with their spin. Ditto for the positrons on Bob's side.
Here's one entry that Alice might have recorded on day 1.
Case 1. She can verify her results by passing all the electrons along a second magnetic field, also aligned with the Z-axis, and she will find all those which had a spin up(+) will still have spin up(+), and all those with spin down(-) will still have spin down(-). This is what is meant by preparing a quantum system in a given state: the first measurement prepared our quantum system in a given state - it forced all the electrons to have either spin up or down along the Z-axis. The second measurement confirmed that. Mathematically, the system is in a eigenstate.
Of course, because the particles are entangled, Bob if he also measures his positrons along the Z-axis, will record for each positron its opposite spin. The law of conservation of angular momentum demands it. Now, there's no mystery here.
Case 2. But what if Bob had chosen to measure along a different axis, say the X-axis, by putting a magnetic field along that axis. Now again he will measure along the X-axis either spin up(+) or spin down(-).
Say the first electron that Alice actually measured to be up (+) on the Z-axis, and would have been down (-) on the Z-axis for Bob's particle is now up (+) on the X-axis, did Bob overcome the Uncertainty Principle, since he now "knows" the z-component, down (-), and x-components up (+), of that particle? Not really. Should he take that particle into a second magnetic field now aligned with the Z-axis, he will find that there will be a 50% chance that it will be up (+), and a 50% chance down (-). So he doesn't "know" the spin along the Z-axis, only along the X-axis. The act of measuring along the X-axis no longer guarantees that he has a particle with spin down (-) on the Z-axis. And this is at the heart of Quantum Mechanics: we don't know the state of the particle until we make a measurement, and whatever the state of a particle was before the measurement, it can be altered if we pass the particle into a different apparatus(Case 2).