but it was about the technical details on observables. Even some of the Internet links confused me...
An observable basically is something you can measure. However compared to just any general measurement, an observable has one specific property: If you repeat the measurement without the system changing in between, you get the same result (this is obviously not true for all possible measurement procedures). Such measurement have indeed been experimentally realized; for example the Stern-Gerlach experiment to measure spin.
That you get the same result for repeated measurements means that after the measurement you have to assume that your system is correctly described by a state for which this measurement gives the observed result (a so-called eigenstate of the observable).
In classical mechanics, this is no problem because there all observables are assumed to have well-defined values all the time, and the measurement only reveals them. However in quantum mechanics, this assumption does not work; it can be shown that this assumption would contradict quantum mechanics.
So a quantum system can be in a quantum state where a given observable has no well-defined value and it is only possible to give probabilities about what value a measurement will give. On the other hand, after the measurement your quantum system has to be described using a state where the given observable does have a well-defined, predictable value. In other words, unless you already had an eigenstate of your observable, the state you have to describe your system with has changed. This change of state is what is sometimes referred to as "collapse of the wave function" or "reduction of the wave function".
Now one of the properties of this change is that it does not follow the same laws as the change of an unobserved system. On the other hand, a measurement is ultimately also just a physical interaction, and therefore one would expect them to follow the same laws as other physical processes. This mismatch is known as the measurement problem. How to solve the measurement problem is what interpretations of quantum mechanics are all about.
Now decoherence is the observation that as soon as your quantum system interacts with other systems (the measurement apparatus, or the environment), it becomes entangled with it, and therefore you cannot give the quantum system a wave function by itself. Instead you find that if you ignore the additional systems in your description, you have to describe the system as if it were a statistical mixture of wave functions. And it turns out it's the same statistical mixture that you also become when you measure your system. Some people claim that this already solves the measurement problem, but others disagree.