MWI for the layperson:
In classical mechanics, each particle has a position and velocity. If there are N particles, the state of a physical system at a given time is given by a list of the positions (a configuration, or point in 3N-dimensional configuration space) and of the velocities for each particle:
X1 = (x1,y1,z1)
classical state: (X1(t), V1(t)), (X2(t),V2(t)), (X3(t),V3(t)), ...
In QM, there is instead the wavefunction, psi, which is a complex-number-valued function on what would classically be configuration space plus the space of spin configurations, and is a function of time:
quantum state: psi(X1,S1, X2,S2, X3,S3, ... , t)
(Spin, Si, takes on a small set of discrete values.)
This is a classic way of generalizing something: instead of a point in a space, there is a function on that space. It must be emphasized that the wavefunction is not a function on regular 3-dimensional space, but on the 3N-dimensional space of configurations. This high-dimensional arena is responsible for many of the counterintuitive properties of quantum mechanics.
If the wavefunction is somewhat sharply peaked near a configuration, though with a wavelength small compared to the width of the peak, it will behave a lot like a classical system; the peak will follow a nearly classical trajectory as a function of time. It is natural to conclude that any interesting things done by such a classical system, such as performing computations, will be done by the wavefunction. It is just like a classical world, only a little 'fuzzy' due to the finite width of the peak. Indeed, roughly this picture is probably how most people think of QM, including chemists - a classical world except that electrons and similar particles are spread out instead of being concentrated at a point.But that is obviously not a complete picture, because the wavefunction is not concentrated around a single peak. Roughly speaking, there are many peaks, representing quite different classical configurations (e.g. the alive or dead configurations of Shrodinger's cat), and many places even away from the peaks where the wavefunction is nonzero. Yet the world we see resembles a classical, single-configuration world. How can we explain that?
There are three main approaches. The first is some modification of QM in which only one peak remains, while the others vanish - this is called 'collapse of the wavefunction'. There are three main problems with this: 1) it introduces a lot of complexity to the model which might be avoided by another approach; 2) it violates things physicists like such as conservation laws; and most importantly 3) it doesn't work because generally speaking, in proposed models that give mathematical details of 'collapse', small residues remain in the other parts of the wavefunction. Small or not, these residues still go through trajectories that should give rise to computations and thus observers - unless we have reason to believe that probability is higher in high amplitude regions; but if we do, we might as well just go with the simpler MWI since deriving that is its main problem.
The second main approach is hidden variables. As we have seen, local hidden variable models are ruled out by Bell's theorem, but nonlocal models exist that don't have that problem - most famously, the Pilot Wave Interpretation (PWI). In the PWI, a classical-like configuration point 'surfs' along the wavefunction. It has been shown that under quite general conditions, the probability distribution for the point evolves in time to match the Born Rule of QM.
Two main problems have been raised for the PWI and similar models. 1) It is nonlocal, and has a preferred reference frame contrary to the spirit of relativity. This is really a matter of taste, and I don't consider it a fatal problem at all, though I do think the nonlocality is an undesirable feature if other models can avoid it. 2) More importantly, it doesn't get rid of the other peaks in the wavefunction at all; it just adds a new trajectory of the hidden configuration point. The wavefunction is still there and should still perform all of the interesting computations as it would in the MWI. Thus, the PWI has been called 'Everett in denial'. Valentini [http://arxiv.org/abs/0811.0810] has denied that charge but his straw-like arguments are easily demolished as Brown has done [http://arxiv.org/abs/0901.1278].
I must note an important exception to the many-worlds property of the PWI: In some versions of what is proposed for quantum gravity, the wavefunction of the universe does not evolve as a function of time; this is known as the Wheeler-DeWitt equation. That would seem at first glance to rule out observers in those models (that remains to be seen even for just a wavefunction). However, the PWI hidden variables would evolve in time even though the wavefunction doesn't, making a single-world model out of it. While interesting, I find it implausible that something as complex as the wavefunction of the universe would have to be in such a model could be an initial condition.Finally, there is the MWI itself, as first proposed by Everett and in various forms by others. In its basic form, this has the simplest mechanics as it just lets the wavefunction evolve over time, adding no hidden variables or collapse-inducing modifications to the dynamics. There are many peaks in the wavefunction which follow various trajectories and implement various computations. [Much more to be said on that rough sketch.] Each individual observer only notices a single classical-like world because that is the one associated with the motion of the peak giving rise to the computations of his own brain; the others don't have any effect on him.
This appealingly simple picture, however, raises a problem of its own: In order for our observations to be at all typical, the Born Rule (which relates probabilities to the square of the wavefunction) must hold, at least to some approximation. This means that small amplitude peaks are less probable than large amplitude peaks. Since the trajectory of a peak (which makes it perform computations and so on) does not depend on amplitude, why would that be the case in the MWI?
The possibility of derivation of the Born Rule in the MWI is the central problem in interpretation of QM. If the Born Rule does follow from the MWI, then the case for the MWI is made beyond a reasonable doubt. I will discuss in other posts various attempts to derive it.
If it does not follow, then the problem remains - what interpretations could work? Continuous collapse models and the PWI would still not work because they would still be the MWI in disguise due to having the wavefunction (with its wrong probabilities) as part of their ontology.
One possibility that looks like it should work in any case is making an honest Many-Worlds version of the PWI: having infinitely many sets of the hidden variables. The simplest version, that of having every point in configuration space sporting a wave-surfing hidden variable, is called Continuum Bohmian Mechanics (CBM). These hidden variable worlds could then outnumber the ones in the wavefunction, producing the Born Rule for typical observers. Of course, this model is more complex than the standard MWI. Also, it still would leave the question of what observers are and how to count them.
Quantum gravity remains an unsolved problem, and the solution may play a role in interpretation of QM, perhaps providing a new set of variables to work with.
Another problem is that in the long term, long after normal observers have died out, spontaneously assembled bits of random matter (which a cosmological constant would produce) would eventually include short-lived observers who would outnumber the normal ones over the history of the universe by perhaps an infinite factor. These Boltzmann Brains, and the necessity of getting rid of them in terms of their effect on typical observations, provide important constraints on what the real answers could possibly be. This deserves a post of its own, at least.
Another (and not unrelated) topic that will get its own post is the Everything Hypothesis, which postulates that every possible thing must exist as an explanation for why things are how they are.