The foundations of quantum mechanics (Section 11.3) of the Physical Chemistry textbook, introduces the idea that electrons must be considered as both a particle and as having wave characteristics. This concept is absolutely key to understanding quantum mechanics and yet is sometimes hard to get a handle on.
Consider for a moment a wave of water. The shape, movement and behavior are both familiar and comfortable. What is the wave made of though?  Could it be very generically said that a water molecule is then acting as both a particle and as a wave?  Is this a bothersome statement?
Let’s consider one water molecule as a wave, granted it does require more than one molecule to make the wave, but it does serve our purpose here. That molecule travels from high points (crests) to low points (troughs) with a fixed height (amplitude).  It has a definite wavelength (crest to crest distance or trough to trough distance) and its energy is obviously seen by its frequency.  Electrons can be seen in the same way.
How would this molecule of water (or an electron) react with others near it?  If they were completely identical it would enhance the wave, if they were different in any way, the two would interfere with each others patterns and eventually break down the wave.  If they were completely opposite, it would obliterate the wave immediately.
This pattern of constructive and destructive interference can be seen with any type of water waves, from subtle ripples created by a thrown rock in a still pond to dramatic changes of crashing waves on seashore.  It is the same with electrons; figure 11.9 (see above) in the physical chemistry text describes the de Broglie wave association.
The locations of the electrons around the nucleus can be simplistically seen as areas of constructive interference (places they do exist) vs. destructive interference (places they do not exist) as described by the electron “orbits” in the Bohr model of the atom.
Could we identify the location of this molecule of water (or an electron) at any given moment?  Does it ever hold still long enough to get its position?  If it is always moving, there is no way to accurately predict its exact location, conversely, if it is holding still (stationary), there is no way to determine its movement (momentum).
This leads us to the uncertainty principle.  It states that the position and the momentum cannot both be known simultaneously.  If one is found, the other becomes meaningless and unreliable.
Does this mean that they do not exist at the same time?  Does the water molecule exist even though it is traveling in a wave?  Of course, but if its location is fixed, its movement is meaningless at that moment because it can’t be moving if we know where it is!
There is a wonderful group of interactive tools in this section to illustrate both the wave motion of the electrons and the uncertainty principle and the mathematics behind this revolutionary look into the behavior of electrons.