Here I will explain some of the terms that seem most likely to cause confusion and which are most directly involved in the interpretation of QM. Rather than technical definitions, here I am concentrating on their role in interpretations. In the next post, definitions Part 2, I'll define some slightly more technical terms which often arise in the context of discussions of QM but which are not as essential for the basic interpretation issues.
wavefunction: The mathematical model of QM, in it most standard formulation, deals with a complex-valued function of the configuration space that undergoes wave-like motions. In a fit of inspired creativity, it was dubbed the wavefunction ;) It is often represented by the Greek letter psi, which looks like a U with a vertical line through the middle and extending below the curve.
measurement: This usually refers to an experiment in which a human observes a macroscopic instrument to determine the outcome. As such, it is an emergent phenomenon and can play no fundamental role in the mathematical model of the physical world. However, it certain plays a fundamental epistemological role in our ability to learn about the world.
When a 'measurement' occurs, the result is one of the allowed results (an eigenvalue of the measurement operator) and subsequently the wavefunction appears to behave as if it had been placed in the corresponding eigenstate (see below).
eigenfunction/eigenvalue: These terms from German, now used in linear algebra, describe mathematical properties of certain functions. Eigen- means characteristic. Each measureable quantity (such as energy) corresponds to a linear operator. Each such operator A give a spectrum of solutions to the equation A psi_i(x,..) = c_i psi_i (x,..) where c_i is a constant. Here i is an index for however many values work for that operator. The different constants are called eigenvalues, and I'll let you guess what the corresponding functions are called ;) A wavefunction that is an eigenfunction is called an eigenstate.
Copenhagen Interpretation: This was "the standard view" of most physicists during much of the 20th century. Essentially, it said that "when a measurement occurs", the wavefunction "collapses" to give one of the allowed outcomes. It was never possible to define the exact circumstances under which a measurement was supposed to occur, or to say what "collapse" was like. This interpretation has become widely recognized as incomplete at best, or less charitably, as ill defined and utterly implausible. In practice, it often meant "shut up and calculate" - it allowed the interpretation question to be swept under the rug so that physicists could work on practical problems instead, like making atomic bombs. It is no longer taken seriously by most philosophers of QM.
Shrodinger equation: This is a deterministic, linear equation that gives the time evolution of the wavefunction. In the MWI, this equation always holds true.
It does not produce any "collapse of the wavefunction" so certain other interpretations must modify it, either explicitly (continuous collapse models) or by hand waving talk about 'measurement' (Copenhagen).
"collapse of the wavefunction": It was long believed that another process, not described by the Shrodinger equation, must occur during 'measurement' in which a single random outcome is chosen - the so-called 'collapse of the wavefunction'. In 1957, Everett argued that no such 'collapse' is needed to explain what we see - he proposed the MWI.
Caution: Even people who believe the MWI sometimes use a sloppy terminology in which they talk about "collapse of the wavefunction" when it is supposed to be understood that they really only mean the illusion of such collapse due to decoherence. I dislike this misleading terminology.
Born Rule: When a 'measurement' occurs, the probability of each outcome is given by the absolute value of the square of the overlap integral of the wavefunction with the corresponding eigenstate. It is an open question as to whether and how the standard (Shrodinger equation only) MWI can explain this, or if not, what does. This is the key issue in interpretation of QM.
linear superposition: One of the most important properties of the Shrodinger equation is that it is linear. This means that if F1 is a solution of the equation, and so is F2, then the sum (superposition) F3 = F1 + F2 is also a solution. (The actual solution that physically occurs depends on the initial conditions - the starting state.)
In a measurement-like situation, and more generally whenever two systems interact, the solution will usually have a branching type of behavior. For example, when a photon hits a half-silvered mirror, there will be part of the wave that is reflected and another part that passes through. The subsequent behavior of these two parts of the wave does not depend on what the other part is doing. If the two parts are brought back together, the resulting wavefunction is a linear superposition of the parts. This can result in an interference pattern.
Linearity forbids collapse because if F1(0) evolves to F1(t), and F2(0) evolves to F2(t), then F1(0) + F2(0) must evolve to F1(t) + F2(t). Collapse, by contrast, would mean replacing the sum by randomly selecting only one, either F1 or F2.
decoherence: This refers to the way in which different branches of the wavefunction stop interfering with each other. Linearity prevents different terms in a superposition from changing each other but it still permits cancellation or reinforement between parts of the functions - interference patterns, e.g. a positive part of F1(x) cancelling a negative region in F2(x).
Decoherence usually means that a system becomes entangled (correlated) with the environment in a robust way. This is generally an irreversible process in the statistical sense, much like an increase in entropy in statistical mechanics.
Once entangled with the environment, interference patterns are no longer seen because the functions F1(x1,x2,...) and F2(x2,x2,...) now have most of their nonnegligable regions in different parts of configuration space: They may still overlap in terms of the x1-dependence (the microscopic system under study), but they occur in different parts of the environment variables' space, e.g. x2.
In principle, an interference pattern could be restored if the x2,... dependence were also brought back into overlap. In practice, there are so many particles in the environment that doing this is not feasible. Thus, decoherence creates the illusion of irreversible 'collapse of the wavefunction".
Also, it is now fairly well understood that in measurement-like situations, in which an interaction exists that tends to seperate out components of the wavefunction that have different eigenvalues for what is being measured, the different eigenstates will tend to decohere. This explains part of the measurement puzzle from the MWI perspective.
entanglement: This means that the wavefunction has correlations between the states of two or more systems. For example, F(x1)G(x2) is a product state and is not entangled, but F1(x1)G1(x2)+F2(x1)G2(x2) is a correlated, entangled state. Entanglement with the environment results in decoherence and the illusion of "collapse of the wavefunction".
Entangled states between small numbers of controlled particles are also important, because they display various non-classical behaviors, such as violations of Bell's inequalities when measured, and are useful in quantum computing and quantum cryptography.
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