I'm not a physics guy, but I've read quite a bit about this stuff. This is the best I can do.
Imagine the particle as a little bar magnet that when fired into a magnetic field either curves up or down, and that this is called its "spin". Because spin is quantized, you can never get half a spin, or a bit of a spin - it's always just UP or DOWN. So let's say we have an apparatus that measures the spin of a particle. The value we get depends on how we orient the apparatus. If we align the apparatus pointing up and measure the spin of a particle that we already know to be spin UP, it will (unsurprisingly) measure the spin as UP. However, if we rotate the apparatus 90 degrees and measure the spin again, rather than measure no spin, it will measure spin UP with 50% chance. So the orientation of the apparatus determines the probability of measuring a spin UP particle as having spin UP. An angle of 60 degrees would have given a probability of 2/3.
Once you've measured the spin of a particle, if you measure it again with the apparatus at the same angle, it will always give the same spin. For example, imagine some random particle happens to measure spin DOWN. Now, without rotating the apparatus, it will always measure the same particle as having spin DOWN. However, if we rotate the apparatus 90 degrees, it will have a 50% chance of measuring spin UP or DOWN - and then all subsequent measurements at that angle will be the same. This means that you can take a particle of unknown spin and simply measure it to "prepare" it with a spin.
Some processes naturally produce a pair of particles whose spins are entangled. That means if one of them is spin UP, the other will be spin DOWN. For example, say you measure particle A, and you have no idea what its spin is in advance (it's random), but it turns out to be spin UP. Now you can say with 100% confidence that if you measure particle B at the same angle it will give spin DOWN. This is true even if the particles are separated by vast distances.
Imagine you generate a stream of entangled pairs, and send one particle from each pair to Alice, who records their spins, and the other particle from each pair to Bob, who does the same. Alice records the sequence UP, DOWN, DOWN, UP, DOWN, UP, UP. Bob records DOWN, UP, UP, DOWN, UP, DOWN, DOWN. Each sequence appears random, and yet they are perfectly anti-correlated.
The simplest way of understanding quantum mechanics IMO is with the many worlds interpretation, which can be thought of as *relativity of state*, and in which an entanglement between two systems (e.g. two particles) is just a *state dependency* between them. Most people are familiar with the idea that motion is relative: If two spaceships are coasting toward each other at constant velocity, the question of which one of them is moving depends on the frame of reference. It's just as true to say that spaceship A is stationary while spaceship B is moving as it is to say the inverse. Both descriptions are equally true.
Imagine a cat in a box. It's a perfect black box, meaning no information can leak in or out until we open it. Within the box, we set up some equipment to measure a particle's spin and then rotate the apparatus 90 degrees and measure it again. If it's UP, the cat dies, otherwise the cat lives. The cat is now entangled with the particle. There is a state dependency between the particle and the cat. Spin UP = dead cat, spin DOWN = living cat. Both are equally true. There are two reference frames. In the spin UP reference frame, the cat is dead, in the spin DOWN reference frame, the cat is alive. Both are true. The cat is in a superposition of states DEAD and ALIVE.
Imagine a scientist opens the box to check on the cat. He decides that if the cat is dead, he will kill himself too. As soon as he opens the box he finds himself in one of the two reference frames and the cat is either alive or dead.
Now let's say the scientist and the black box containing the cat are themselves contained within a larger black box. From our perspective outside the box, the scientist too is in a superposition of dead and alive. There are two reference frames: the spin UP frame and the spin DOWN frame. In the spin UP frame, the cat and the scientist are dead, and in the spin DOWN frame they're both alive. Both are equally true. The scientist, the cat, and the particle are all entangled, meaning there is a state dependency between them.
Now we open the box. It's a 50% chance we'll see a dead cat and a dead scientist. Let's imagine we get lucky and happen to find ourselves in the spin DOWN reference frame in which the cat and scientist are still alive. Now imagine all this has taken place within an even larger black box. To any observer outside of that box, *we* are now entangled with the cat and the scientist and are in a superposition of two states: one in which we observed a dead cat and a dead scientist, and one in which we didn't. So if we are in a superposition, why doesn't it feel like it? Because there are two copies of us.
These black boxes do sort of exist in reality. Every time some quantum event happens (which is basically everything at the atomic scale), you can imagine a sphere of influence expanding out at the speed of light. When that sphere engulfs you, you will (subjectively) find yourself inside one particular reference frame - but there will be another copy of you in the other reference frame(s).
That's basically my summary as a complete layman. I expect I've got loads of stuff wrong.