Yikes! 3 answers and no help! Glad I found this.
The books you're reading should be explaining these things to you. Maybe you're going too fast? Anyway, I can answer your question, but you're going to have other questions. It's a difficult subject and you might want to find other people who can help you. I recommend physicsforums.com .
You're absolutely right to be shocked and confused that a particle and a wave are the same thing, and you're asking all the right questions. The early discoverers of quantum mechanics were also shocked and confused. After a lot of experiment and study, they figured out the key: the wave contains information about the probability of finding the particle in a particular place and time.
Think of an ordinary wave like an electromagnetic wave propagating out into a room from a light bulb or something. Now imagine that we turn the light on and off very quickly. So that light wave that propagates outward is a spherical pulse that expands outward and eventually hits the walls and gets absorbed or something. That spherical pulse is certainly a wave, but it has a specific size (it's a sphere with some radius and also some thickness representing how long the light was turned on) and also contains a specific amount of energy (the total amount that the light bulb sent out while it was on).
Now, if we make the light bulb dimmer and do the same thing, the pulse has the same size as before, but less energy. It has less energy because the amplitude of the wave is smaller; it's a gentler ripple in the electromagnetic field than before. The amazing thing that QM says is that if we keep making it dimmer, eventually we find a MINIMUM amount of energy that the pulse can have without being zero! If we try to make it dimmer than that, the electrons in the bulb simply refuse to radiate any of their energy as a light wave. Not only that, but we find that the amount of energy in the pulse can only ever be a multiple of this minimum amount! So the dimness/brightness of the light pulse always goes up or down in tiny steps. The size of a single step of that minimum energy in the wave is called a "quantum".
So we see that our spherical pulse is always made up of some number of quanta (plural of "quantum") on top of each other. Each quantum can be a sphere in the same shape and size as the whole pulse they make up (actually, they won't really be for a light bulb, but that's just because a light bulb is an awkward and uneven light source, so ignore that detail). A quantum of the electromagnetic wave is called a "photon".
But wait a minute! You've heard of photons and you've heard they're supposed to be "particles". I just said a photon can be an expanding sphere with some thickness, which is obviously not a "particle" in any normal sense. Well here's the trick: when that minimum-energy wave hits an electron, it can't use a little piece of its energy to wiggle the electron, as we would expect from the usual wave picture. Why? Because then what's left of the wave would have slightly less energy, but it already had the minimum possible nonzero amount. So, we find that ALL the energy of the wave must go to wiggle ONE electron. This is the really-insane-but-true part.
When that big spherical photon hits an electron, it must choose (randomly) to either be COMPLETELY absorbed by that electron, or not absorbed at all by it. If it decides not to be absorbed, it keeps expanding outward and faces that same choice again at the next electron it hits. Eventually, it will be absorbed by one of them, and when this happens, the whole wave, all of it, even if it's a mile in diameter by now, instantly ceases to exist. The electron then wiggles as if the full energy of that spherical wave were concentrated right at that one point.
Another way of saying this is that the wave is actually a probability wave. The intensity of the wave at a point in space tells you the probability density of the photon being received by a hypothetical single-electron-based photon detector placed right at that point.
Similarly, an electron is just a quantum of a different kind of wave in a different kind of field (the electron field, not to be confused with the electromagnetic field). So the intensity of the electron's wave function can be viewed as the probability density of "finding" an electron at that point if you look for one there. However, until you actually look for it somewhere, it really is in all the points in that wave simultaneously. It's not until you "measure" it (interact with it somehow) that the wave decides either to "collapse" to a point particle you can measure, or to keep going.
Hope that helps.