What is a wave function in simple language?

A wave function is a complex-valued function $f$ defined on ${\mathbb R}^1$ (if your electron is confined to a line) or on ${\mathbb R}^2$ (if your electron is confined to a plane) or ${\mathbb R}^3$ (if your electron ranges over three-space), and satisfying $$\int |f|^2=1$$ (where the integral is defined over the entire line or plane or 3-space).

Every electron has an associated wave function, and any function satisfying the above can be the wave function associated to some electron.

The wave function tells you everything there is to know about the electron. For example, if $A$ is any set, and if you perform an experiment that answers the question "is the electron in the set $A$?", then the probability you'll get a "yes" answer is given by $$\int_A |f|^2$$

(So in particular, if $A$ is the entire space, you're asking "Is the electron anywhere at all?", and the probability of a yes answer is $1$.)

The next steps are to learn:

1) How do I use this wave function to predict the outcomes of questions about something other than the electron's location, such as, for example, its momentum?


2) How does this wave function change over time?

I don't think you're quite yet at the point of addressing those questions (though you will be soon enough).

The wave function is the solution to the Schrödinger equation, given your experimental situation. With a classical system and Newton's equation, you would obtain a trajectory, showing the path something would follow: the equations of motion. For a quantum mechanical system you get a wave function, and the rules it obeys over time. With this you can determine the odds for your particle to be someplace, which is as close as you can get to a trajectory.

You normally learn most of the math first, then Newtonian mechanics, and then quantum mechanics. Hopefully by then you will be able to put it in context.

A "wave function" is a mathematical model (or representation) of a given wave. A "function" is represented by the symbol $f$. It can be a function of distance (x), time (t), space (r), etc. and is usually represented by an equation. If the equation represents a wave, then the function is a wave function.

For example, a simple wave with constant amplitude and varying in time can be described by: $A \sin{t} $. Its wave function would be $f_{(t)} = A \sin{t}$. You can evaluate it over some interval, by integrating over the interval.
In the case of the electron, its wave function has already been described in WillO's answer.