(Sorry for the delay in answering; Sunday night movie with the wife. :-) )
The term "superposition" means, in this context, two things--or rather, one thing, but expressed two ways:
(1) Given a particular physical setup--the collections of forces (or, equivalently, sources of potential energy), both internal and external, that act on a quantum particle, along with the initial conditions of the system--quantum theory cannot produce a single answer to any question you might pose, but only a list of possible answers, along with the probabilities that a measurement of the relevant physical quantity will produce each possible result.
For example, if I ask "In my particular experiment, what is the magnitude of the orbital angular momentum of the electron in a hydrogen atom", quantum theory will produce a list of (say) 5 possible values, along with the probabilities of obtaining the 5 values: 2%, 10%, 76% 10%, 2%--when you make the measurement. Thus, after the measurement, the angular momentum has a definite value; but before the measurement, the most we can say is that the electron will be found in one of those states, according to the weighted probabilities.
But, that is a lot of words; so, the phrase "quantum superposition" was invented to mean all of that. The common phraseology is to say that "prior to measurement, the electron is in a superposition of these 5 quantum states".
(2) The math way to say exactly the same thing is the state function (i.e., the solution to Schrodinger's equation for the given potential energy function) is a function that is a superposition (a sum) of so-called "basis functions" (or "basis states"); each basis state is one of the 5 states mentioned above.
This is what people mean when they write things like "the electron can be in two places at the same time", but it is a horribly imprecise and misleading way to phrase it. --But I understand why writers do it; look how many words it took me; what newspaper editors would allow 400 accurate words when 40 semi-accurate words will sort of do, and who the hell besides a few physicists will care, or even know?