I can give you the axiom of quantum mechanics and hope that's what you're looking for:
The state of the particle is just whatever the particle is doing. Let's say the particle is sitting behind your couch; then it's in the "behind-the-couch state." If it's on the ceiling, it's in the "on-the-ceiling state."
To find out something about the particle we ask it questions, such as, "Where are you?" In other words, we make measurements, but we can't measure the wave directly. The information to answer the question is in the particle's state. Grab hold of this axiom: Quantum mechanics says there are only certain allowable answers to our questions.
Each allowable answer questions corresponds with the particle's state. For example, if we ask the particle where it is and it answers, "I'm behind the couch," we say it's in the "behind-the-couch" state. The answer, "I'm on the ceiling," corresponds to the "I'm-on-the ceiling" state. States are purely states of position.
If particles were always in pure states, quantum mechanics would be easy. The "behind-the-couch" particle could always reply, when we ask where it is, "I'm behind the couch." Unfortunately, particles are rarely that simple. Often they exist in mixed states.
For example, the particle could be in a state that combines both the behind-the-couch and on-the-ceiling states. It could be in a state that's 30 percent couch and 70 percent ceiling.
But a problem arises when we ask the particle where it is. It cannot say, "I'm 30 percent of the way between the couch and the ceiling" because it is not somewhere in the middle. Rather, it is in both places at once. Think again about what the wave looks like. If the particle were in a pure couch state, its wave would be flat everywhere in the room except for the wiggly part behind the couch. If the particle were in a pure ceiling state, its wave would only wiggle on the ceiling. But since it's in a mixed state, we have a wiggly part behind the couch and a wiggly part on the ceiling. There's still just one particle, but it's smeared out in two wiggly parts. It's in two places at once.
If you lived in Quantumland, you wouldn't have to choose where to sit in a movie theater-front, middle, or back. You could sit in all three places at once.
