What is meant by the "probability density" of finding an electron?

So imagine you have a friend who's an alcoholic, and your phone is broken. You know he's not home, so he's probably at his favorite bar. If you had an aerial map of the area and had to guess where he was, you'd draw a dot on his favorite bar stool [most likely place he is], a circle around the bar building [still could be here], and a dotted line encompassing all of the shrubs, taco truck, and parking lot around the bar [he MIGHT be getting a taco, beat up in the parking lot, or passed out in the bushes].

So in this little parable, your friend is an electron and you have drawn a probability density.

Extra credit: the reason we have to use a probability density is that, for complex mathematical reasons [Heisenberg uncertainty principle] you can't know your friend/electron's velocity AND position, so you're making a mathematical guess about where your friend/electron is, and then describing that mathematical guess in terms that are confusing to us plebes. But really it's just a heat map. Also, depending on what element/number of electrons/orbital you're dealing with, you can get different shapes. Don't be confused - it's just like knowing more friends each with their own different bar, and they don't like each other so they don't overlap.

In quantum mechanics, we clearly know that electron isboth a wave and a particle.Due to this characteristic feature of the electron, being a wave it can exist at numerous places at an instant. So the probability distribution function in relevance with the Schrödinger wave equation are the chances of finding an electron in the region which needs to be studied.

In quantum mechanics we often need to find the probability at some position $x$. However, $x$ is a continuous variable with an infinite number of values so it does not make sense to ask what is the probability of being at some exact position , say $x=1/4$, as this will be vanishingly small. Instead we imagine that the probability is that of being at position x to $x + mathrm{d}x$ where $mathrm{d}x$ has some very small value. If $f(x),mathrm{d}x$ is that probability then $f(x)$ is the probability density which must be a real non-negative number. It turns out that quantum mechanics postulates that the probability density is given in terms of the wavefunction is $|psi|^2$.