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: