Disclaimer. I am not a physicist. Please step in, write a better writeup, and /msg me to have this clumsy attempt nuked out of orbit. Thanks.
At the quantum level, objects (elementary particles) are so tiny, they can no longer be said to have a definite position, velocity, mass, and energy; the world is described more accurately, at this level, in terms of wave functions that correlate these entities. These functions can be interpreted as describing the probability of an elementary particle having a particular position, velocity, mass, energy at a particular time.
So think of the world at the quantum level as being described by a gigantic system of equations, each representing the continuum of possible states of a single elementary particle. The effect of a measurement, the observation of a particle, is to fill in some of the values in this equation, which effectively results in a new system of equations. The first system of equations predicted the exact outcome of the measurement, in the sense that it predicted a continuum of possible outcomes ("possible realities"), with the relative probability for each of these to occur. The actual observation has 'proved' one of these possibilities to be reality. Mathematically, it simplifies the system of equations; those equations now describe all possible outcomes of any further possible measurements consistent with the measurement that just took place (and all the measurements before it, if any).
So we can think of any such system as representing a mindblowing number of possible combinations of position, velocity, mass, and energy of elementary particles. Any such combination can be thought of as a 'possible world'; if all possible measurements could actually be done, one of them would emerge as the actual world we live in.
This description is known as the many worlds interpretation of quantum physics. It is an attempt to formulate in natural language what is really a mathematical property of the mathematical constructs used to describe the world at the level of elementary particles.
Such interpretation has no meaning in physics. The substance of quantum physics is formed by mathematical models that relate the probabilities of the outcomes of measurements; their validity is assessed by checking if actual measurements in the real world correspond to the predictions made by the model. The business of physics is to propose, refine and verify such models, in order to provide more and more accurate predictions of the measurements made in reality. Calling a particular mathematical construct within the model a world, or a possible world, takes the discussion outside the realm of science, and into metaphysics, which, to scientists, can be described as the study of how we conceive of the world. Put another way, it is a study of what we mean exactly when we use the words 'possible' and 'world'.
Modern physics doesn't study words, it studies equations.