What is a wavefunction?
Let's put aside the whole "collapse" issue -- not all physicists agree that it is a sensible concept. NM's comment has a link to the Measurement Problem, and I'm not a good enough theorist or philsopher to contribute to that argument.
What is the wavefunction?
- "Is wavefunction only a convenient way to say it's located somewhere close to here, but we're not sure exactly where until we measure it?"
- "At any particular point in time/space the object is in a definite spot with a definite set of properties, but we can only make a reasonable guess?"
How do I get to that point? Well, quantum mechanics is one example of a "field theory" (electromagnetism is the most familiar classical field theory). The equations we write down (the Schrödinger equation non-relativisitically, the relativistic Dirac and Klein-Gordon equations) to describe how quanta behave are coupled partial differential equations (PDEs), which relate the values (and derivatives) of the field at every point in space to their evolution in time.
A PDE which relates the time and spatial properties of a function is either a wave equation (if the solutions are sines and cosines) or a diffusion equation (if the solutions are exponentials). The Schrödinger equation is a wave equation, and we call the solutions wavefunctions. Electromagnetism also has a wave equation, which is how we get radio, light, etc.
The difference is that the functions in EM are "real valued:" the value of the field at each point in space/time is a regular floating-point number (the "phase" in EM is determined by the relative values of the field and nearby points). The wavefunction is a '''complex valued''' field -- at each point in space/time, the field has both an amplitude and a phase (or equivalently a real and an imaginary component). This means that wavefunctions can interfere in ways more complex than simply "adding" or "subtracting", which can have quite interesting consequences.
You get probabilities by taking the square (norm) of the wavefunction. This procedure gives you a real value, a probability, at each coordinate. When you make a measurement, those probabilities determine which coordinate value you see as the "location" of the quantum. The actual result is random, but that isn't because "we're not sure exactly." The quantum objective does not have a single coordinate location until we make the measurement.
How that happens, whether by "collapse," "decoherence," "many worlds splitting" or something else, is a subject of intense philosophical and experimental argument.