Why oh why
Why are people afraid of saying what "superposition" is?
It's just a vector of length 1, with complex-valued coefficients.
It lies on the unit sphere of an N-dimensional vector space.
Each dimension of that vector space represents the state of the system.
If the vector lies perfectly aligned with one of the axis, the system is in a definite state. Otherwise it is in a superposition.
For example, the qubit can be in states 0 or 1.
This gives a 2-dimensional complex vector space, with the state vector on a "circle" (actually a 4-d real sphere)
A system with 2 qubits can be in states 00 01 10 or 11.
This gives a 4-dimensional complex vector space, with the state vector on a 4-D complex sphere.
The quantum computer rotates that vector according to some linear differential equation.
For "observation", in the simplest case, you take the coordinates of the vector along each of the principal axes for 00, 01, 10, 11 (this is a complex number). Take the length of the complex number, square it. This is the classical probability of "observing" the system to be in respective classical state 00, 01, 10, 11.
Given that you want to solve a problem, you will set things up so that one of the axes is solution that you seek (evidently, it corresponds to a bitpattern). So you want your quantum computer to follow the linear differential equation that gives you the correct bitpattern with reasonable good probability upon "observation".
And this is actually an extension of classical probability theory, it's just that nature apparently likes complex numbers: working with real-valued probabilities and state vectors moving on planes in an N-dimensional real state space is tired.
Now, what kind of algorithms can be build with this extended probability calculus?
Fast factorization. Simulation of quantum systems. Fast lookup in databases. Any other.. ?
How to solve the "find the differential equation for my problem"? How to map the differential equation to the architecture described in the article? There is work to do!!