What do you mean, a wave? An example is in order. Suppose a billiard ball is bouncing around a table. At any time it is in precisely one place and thus its probability of being at any point on the table is either zero (for all the places where the ball isn't) or one (for the place where the ball is). If we confine an electron to a small rectangle, it doesn't act the same way. The electron has a certain probability of being at any point of the table. The probability is not simply one at one point and zero everywhere else. "But wait," you say, "when I look for the electron, I'll find it somewhere so it's there with probability one and everywhere else with probability zero!" Well, it would be pretty to think so. However, if you do experiments with electrons, one finds that the probability of finding the electron at a particular location acts like a wave.
One experiment to test this is called a "double slit" experiment. A beam of electrons is directed at a screen with two slits in it. Behind the screen there is a detector which records the position at which the electron hits. If the electrons act like billiard balls, you will see two big peaks on your detector, one behind each slit. When the experiment is performed, several peaks are seen on the detector as well as several places where no electrons hit. This is difficult to explain from the billiard ball point of view. You might try to rescue things by claiming that if a few billiard balls go through at the same time, perhaps the hit each other and cause these strange patterns. Unfortunately, even if you only send in one electron at a time, you get this strange pattern on your detector (called an "interference pattern"). Still, it's not clear that the electron is behaving so strangely, maybe you screwed up the experiment and the electron is getting pushed around as it flies toward the detector. However, if you close one slit, you get the pattern you would have expected at the other slit. So, with only one open slit the electrons do exactly what you expect but once you open the second slit you don't get that pattern behind each slit, you get a new pattern altogether.
This should worry you. After all, we send in one electron at a time, we checked that it does what we expect through each slit alone so how can opening another slit change things? This is a good question to sit with for a while. Somehow the electron "sees" the slit it doesn't go through. Actaully, the electron goes through both slits because it is a wave, not a particle. Imagine a water wave going through our two slits. With only one slit open water hits mostly behind that slit. With both slits open, water goes through and ups and downs in the wave can interfere with each other, causing strange patterns on the detector.
So electrons act like water waves. There's only one problem. When you measure the position of the electron you can't detect the wave. The electron will be measured at only one point, unlike the water wave. This is a deep thing. Quantum mecahnical particles (which is to say, empirically all particles) act like waves when they aren't being measured and then act like billiard balls when they are measured. I'm with you in asking, "how do they know?"
Scattering Theory As I mentioned above, wavelike interference is crucial for understanding certain devices. Recently, devices have been built which actually require one to understand not only the wavelike nature of the electron's scattering but the practical limits on that behavior. Let's go back to the water waves.
Imagine our water wave double-slit experiment. Now imagine we toss a rock into the water between the slits and the detector. If we toss enough rocks or one big enough rock, we'll destroy the interference pattern. Why? Well, the pattern depends on a delicate balance of the wave coming from each slit. If you mess things up enough, the phase (up and downs) of the wave don't work out right and the interference is destroyed. We say the waves from each slit have become "incoherent." In a real device, there are many sorts of rocks. Other electrons are a sort of rock and other objects (impurities in the semiconductor) also cause incoherent scattering.
It is a wonderful achievement of experimental physics that devices can be built where coherent behavior of electrons can be observed. The details are technical but suffice to say that so many incoherent things happen to an electron in a metal or a semiconductor that it is a phenomenal achievement to get observable wave-like behavior of the sort observed in so-called "ballistic heterostructures." One very interesting consequence of that achievement is that it is now imaginable to explore how that coherence breaks down in these devices.
One nice example of this is a device where the electrons bounce many times in the same space, for example off of two walls opposite each other. The first observation you make about such a device is that you can see some powerful interference effects because the many bounces can all interfere. However, you do not see what you would expect if the electrons bounced coherently in the device forever because after a few bounces the coherence is lost due to incoherent processes in the device. I am particularly interested in how such incoherent may be added into a scattering calculation which, at least at first, seems poorly suited to handling anything but wavelike behavior.