Quantum mechanics works. That's the first and most important thing to say. It is the most precisely confirmed physical theory in history, its predictions have been tested to one part in a trillion and held up. Every transistor, laser, and MRI machine is applied quantum mechanics. If you want to talk about what the universe is actually like at small scales, quantum mechanics is your best guide by a very large margin.
And yet physicists and philosophers have been arguing since the 1920s about what quantum mechanics actually means, what it tells us about reality. Not whether it works. Whether it describes something real, and if so, what.
Here is the problem in its simplest form. Quantum mechanics describes physical systems using a mathematical object called a wave function. Before you measure a quantum system, the wave function can represent a superposition of multiple states, an electron that is simultaneously spin-up and spin-down, a photon that is simultaneously taking two different paths. The wave function evolves smoothly and deterministically according to the Schrödinger equation.
But when you make a measurement, something strange happens. You never observe a superposition. You always get a single, definite result, spin-up or spin-down, never both. The wave function appears to "collapse" to a single value. The superposition disappears.
The puzzle: nowhere in the mathematics of quantum mechanics does collapse appear. The Schrödinger equation describes smooth, deterministic, linear evolution, it never produces a collapse. Collapse is added by hand as a separate postulate: "when you measure, the wave function collapses." But what counts as a measurement? When exactly does collapse happen? Is it when the particle hits a detector? When a human looks at the result? When the information is recorded? This is the measurement problem, and it is not an engineering problem, it is a philosophical problem about what kind of thing a wave function is and what the theory is actually telling us about reality.
Niels Bohr's Copenhagen interpretation, the standard textbook answer for 80 years, effectively refused to answer this question. Quantum mechanics describes measurement outcomes; asking what's happening between measurements is meaningless. The wave function is a tool for predicting probabilities, not a description of reality. This answer is pragmatically satisfying and philosophically evasive in equal measure. It works fine if you just want to calculate. It completely fails if you want to know what's going on.
Hugh Everett III, a graduate student at Princeton in 1957, proposed a genuinely radical alternative. His answer: nothing collapses. The wave function always evolves according to the Schrödinger equation, always, without exception. When a measurement occurs, the measuring device enters a superposition of recording "spin-up" and "spin-down." When a human looks at the device, the human enters a superposition of seeing spin-up and seeing spin-down. The superposition just gets larger. It never collapses. Every outcome happens, in a different branch of a constantly branching universal wave function. This is the Many-Worlds Interpretation.
Quick reflection
The Copenhagen interpretation says asking what's happening to a quantum system between measurements is a meaningless question — quantum mechanics only tells you about measurement outcomes. Does this satisfy you philosophically, or does it feel like a refusal to answer rather than an answer? What would it mean for a physical theory to be complete while refusing to describe reality between observations?