Superposition is nothing more than addition for waves. Let's say we have two sets of waves that overlap in space and time. At any given point, a trough may line up with a peak, their peaks may line up, or anything in between. Superposition tells us how to add up these waves so that the result reconstructs the patterns that we observe in nature.
To do this we need to know some basic facts about the waves: their amplitude (how big the wave is), their speed (how fast the peak of the wave travels), their wavelength (the distance between two peaks), and a period (the time it takes for a single position to experience a peak, a trough, and another a peak). There is also an important property that relates the two waves to each other: phase, which measures the time between peaks of the waves.
To create a superposition, we perform an addition that takes into account all of these factors. In some places the waves disappear if a peak from one coincides with a trough of the other. At other locations, two peaks or troughs will coincide to give a much larger wave. This behavior is called interference and it can only occur when adding through superposition is valid.
Sometimes, however, our instruments can't perceive the patterns that are due to superposition, or some of the wave properties change rapidly in either time or space. In this case, you don't have sufficient knowledge to construct a superposition, And, even if you did, the initial superposition would rapidly lose touch with experimental reality due to a lack of coherence, so the only thing you really measure is the total amount of energy. In practice this means we take the amplitudes of the waves, square them and add them up. Since the amplitudes squared are all positive quantities, the interference disappears.
Quantum superposition is handled identically to water waves, but the physical quantity we are talking about is different. Here, the wave is a wave of probability—what is the chance of finding a particle right here?
These waves have all the same properties of water waves, but the amplitude, or height of the wave, is a probability amplitude. This amplitude can be negative, which probably seems a little strange. But, any measurement we care to make only measures the square of the amplitude—something that is always positive.
A quantum superposition has unexpected consequences though. For instance, it is possible for a particle to be in a superposition of position states: that is, there is some probability of the particle being at one location and another probability for the particle being at an entirely different location. Once we make the measurement, the particle will be at only one location. This is called the collapse of its wave function.
Some measurement can make superposition very interesting. Say we can have anything up to ten photons bouncing back and forth between two mirrors. We cannot ask "how many photons are in the cavity?" But, we can ask yes-or-no questions like "are there n photons in the cavity?"
Before we measure there is a superposition state of up to ten photons in the cavity. We ask "Are there three photons in the cavity?" and get a no. The superposition is still in place, but the contribution to the superposition from "three photons" is removed, so all the relative probabilities of the other numbers increases. The wave function has collapsed, but not entirely. The richness of quantum superposition mixed with measurement make for a very exciting set of physics experiments.