A tutorial explanation of decoherence? Is there a tutorial explanation as to how decoherence transforms a wavefunction (with a superposition of possible observable values) into a set of well-defined specific "classical" observable values without the concept of the wavefunction undergoing "collapse"?
I mean an explanation which is less technical than that decoherence is 

the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the trace, with respect to any environmental basis, of the density matrix of the combined system and its environment [...] (Wikipedia).

 A: The Copenhagen interpretation consists of two parts, unitary evolution (in which no information is lost) and measurement (in which information is lost). Decoherence gives an explanation of why information appears to be lost when it in actuality is not. "The decay of the off-diagonal elements of the wave function" is the process of turning a superposition:
$$\sqrt{{1}/{3}} |{\uparrow}\rangle + \sqrt{{2}/{3}} |{\downarrow}\rangle$$ into a probabilistic mixture of the state $|\uparrow\rangle$ with probability $1/3$, and the state $|\downarrow\rangle$ with probability $2/3$. You get the probabilistic mixture when the off-diagonal elements go to 0; if they just go partway towards 0, you get a mixed quantum state which is best represented as a density matrix. 
This description of decoherence is basis dependent. That is, you need to write the density matrix in some basis, and then decoherence is the process of reducing the off-diagonal elements in that basis. How do you decide which basis? What you have to do is look at the interaction of the system with its environment. Quite often (not always), this interaction has a preferred basis, and the effect of the interaction on the system can be represented by multiplying the off-diagonal elements in the preferred basis by some constant. The information contained in the off-diagonal elements does not actually go away (as it would in the Copenhagen interpretation) but gets taken into the environment, where it is experimentally difficult or impossible to recover.
A: The Wikipedia article is rather a lengthy description of decoherence which undoubtedly also prompts this question. Part of the reason is that decoherence is not, in a certain sense, a truly fundamental mechanism amending quantum theory, but an applied mechanism introduced to "explain" collapse using ideas from Statistical Mechanics (those density matrices and concepts of "bath"/environment) supplementing those of quantum mechanics.
Borrowing a key equation from the Tutorial mentioned by Lubos, one writes:
$H = H_c + H_e + H_i$
for the Hamiltonian $H$ understood from a decoherence perspective: its ingredients are $H_c$ which is the simpler component (= system under study) Hamiltonian discussed in elementary textbooks; $H_e$ is the environment and $H_i$ the Interaction (including the experiment).
So this is saying "dont just look at the system alone; look at the entire environment+system" from a conceptual perspective. Abstractly there is no collapse here; although thermodynamic-like arguments are then used to give the (local) appearance of collapse.
To follow this up a Tutorial can do two things:
(1) Explain the thermodynamic aspect a bit more;
(2) Explain what happens to "collapse" at the $H$ level.
Hopefully this helps a little.
A: Dear Nigel, the Wikipedia explanation is not too technical, but let me try to offer an even less technical one:
In ordinary quantum mechanics, the wave function remembers lots of complex numbers - the so-called probability amplitudes. These numbers may be added and combined in many ways and the absolute values of the squares of these sums are interpreted as probabilities.
This is different from classical (non-quantum) physics which contains no complex numbers "beneath" the probabilities.
Probabilistic statistical physics
However, there exists a statistical formulation of classical physics that does deal with probabilities, but ordinary "classical ones". There are no complex numbers beneath them. There simply exists a set of possibilities - e.g. that you throw dice and you get 1,2,3,4,5, or 6 - and classical physics with an uncertain, probabilistically given initial state, may calculate that each of these final outcomes has probability of 1/6.
In this probabilistic classical physics, one doesn't claim to know the positions and momenta $p,x$ of all particles. Instead, one works with the probabilistic distribution function $\rho(x_i,p_j)$ that is a function of all the possible positions and momenta of all the particles. The function describes the probability density that the particles are located in a small volume around a given point $(x_i,p_i)$ of the phase space.
It's the latter form of classical physics that quantum mechanics reduces to after decoherence. Decoherence produces a preferred set of possibilities that may be measured and it calculates the probabilities of each of them, much like you would in the probabilistic version of classical physics. But of course, the underlying theory is still quantum theory and it can only make probabilistic predictions. So decoherence will never actually find a way to decide which of the outcomes is realized.
In other words, there is no "collapse" into a single outcome, just like there is no collapse in the probabilistic classical physics. The density matrix, obtained by tracing the wave function, is not an "objective state of reality": there exists no "objective state of reality". Much like the classical distribution function, it's just a probability distribution describing our incomplete state of knowledge.
The role of decoherence is to choose a preferred set of outcomes that can be measured - e.g. "alive cat" and "dead cat" - and explain why all the other linear superpositions of the preferred outcomes are illegitimate. This result "bans" further interference and other typically quantum phenomena. That allows us to think about the evolution using the classical intuition. But it's still true that the evolution is indeterministic - and it will always be.
A review of decoherence
For others, a more technical explanation of decoherence is e.g. here:

http://www.karlin.mff.cuni.cz/~motl/entan-interpret.pdf

