I am taking seriously your request for a simple language answer, & would also be happy to provide a more complex answer. My answer hopefully provides, by a progressive series of simple statements and examples, a way of actually conceiving what a wave function is. You may want to scroll to the final paragraph.
A function is a mathematical representation of something.
A wave function is a mathematical representation of a wave.
Consider y = sin x
This might describe the momentary snapshot of a wave made by wiggling a rope.
This shape would move along the x-axis with time, so y= Sin(x-t) would be a very simple example of a sine wave moving along a rope.
I'm sure I will be dissed for answering your question literally.
I do have 4 years of tertiary maths and 3 years of tertiary physics behind me, but you did seem to want a very simple example.
An electron is sometimes represented as somewhat like a short burst of waves, and the complex formula for this "shape" is called the wave function. Your quote from the textbook is not a definition of a wave function, it is telling you something about the wave function representing an electron.
You can imagine a straight line with a burst of wave activity somewhere along its length. If the line represents distance, then the position of the burst is the location of the electron. If the line represents time then it indicates when the electron was at some location. The wave function of an electron describes the time, location, and in fact everything we could know about the electron.
Because any method of looking at the electron has an effect on it (rather like using a torch to find out what rabbits do in the dark), we never really know exactly every value of every parameter associated with the electron.
The wave equation therefore can not simply tell us where the electron is, or how fast it is going. It gives us the probabilities of the electron (or other phenomenon) being at any location at any speed.
In our everyday experience, the way we imagine particles to be is quite adequate. It is easy to forget that this conception of a particle is a model, not a reality. In fact there is no such thing as a distinct surface of a particle. When we are dealing with things at a sub-atomic level, our everyday concept breaks down, and we admit that we only know probabilities of the where-and-when of a particle.
The wave function for an electron therefore includes everything we know about the electron, which includes that it is not a particle in the everyday sense, but is an array of greater and lesser probabilities of being somewhere at some time. The complexity is not a reflection of reality, but rather of how difficult it is to create a model than can be understood by a mind that generally needs to relate to what can be perceived by the senses.