It looks like you're referring to the Everetts many worlds interpretation of Quantum Mechanics. This is a solution of the wavefunction collapse, where one can say a measurement has been made. This possibly requires a little elaboration.
First, measurement is not just what a physicist does when he measures the momentum of an electon or an atom, but what every system does when it reacts to another: One could say here, that the first system 'measures' a property of another system and reacts accordingly; but it is also true that the second system measures the first system and also reacts accordingly.
Secondly, and putting it simply, one could say - at least intuitively - that the wave-function represents the possibilities which hold until a measurement occurs, and then the wave-function 'collapses'. One ought to note that the wave function evolution is deterministic until collapse, and this collapse is non-deterministic; and so a certain value is chosen at random for the measured property.
Classically, since Newton, physicists have expected that the universe to be deterministic and real and this held upto the discovery of General Relativity by Einstein. Quantum Mechanic broke this paradigm it seemed irretrievably and the locus of the problem seemed to be the non-deterministic (ie random) evolution of the wave function collapse and also the interpretation of the wave function as possibitities.
It was Everetts aim to retrieve this deterministic & real character for the then new Quantum Mechanics. He posited then every collapse engenders a new universe. Thus a possibility is no longer a possibility but another dimension of reality. This seems a rather high cost for determinism and realism.
As an interpretation it is intriguing but esoteric, and at least physically one requires something more; does this picture of reality provide us e*explanatory power* - Everett attempted to provide one by deriving the basic Borns Rule from it. There is no consensus as to whether this has been done.
Its worth pointing out, given the eye-catching, media and science-fiction friendly parallel world vista of Everetts that a different and much less well known intepretation, Bohmian Mechanics also retrieves realism and determinism by allowing non-locality - that is faster-than-light signalling.
Now, given the tremendous success of the atomic hypothesis in modern physics, and this covers not just the idea of the classical atom, but also quanta (for are they not discrete?) it seemed only natural to think that perhaps even the very structure of space & time is atomic (there are various research programmes that look at this - spin foam and causal nets), and one expects this structure to appear at the Planck scale.
Then given that there has only a finite time has passed since the creation of the universe (the Big Bang), it appears only a finite number of universes are possible, though their number is increasing exponentially.
Its intriguing to consider what kinds of conditions might want to consider that allows either a countably infinite number of universe, or simply (!) uncountable. Personally, my intuition would be that at least one of the basic categories of physical materialism - matter/energy, space-time and gauge-forces are infinitely divisible.
But one also should consider that a rule-of-thumb operates in Physics, which is that infinities are to be avoided: one does not have infinite energies, nor an infinite past, and nor an infinite amount of matter, and nor even an infinite expanse of space. On that basis one might want to rule out an infinite number (of whatever cardinality) of worlds.