I'm with Phinds on this one. I think it's good to talk about what exactly we mean when we say something is a particle, a wave, or has those properties.
Think about a particle. What properties does a particle have? Well, it has rest mass, charge, spin, and acts a certain way when you do things like apply a force to it. You can stash it in a shoebox and come back later for it without worrying about it disappearing. (Unless you left the lid off)
What about a wave? A wave passing through a medium is the oscillation of the particles making up that medium. An EM wave is the oscillation of the electric and magnetic field vectors. BUT, that doesn't mean we can't talk the wave as a whole "thing". It turns out that these waves act according to certain rules. One of them being interference as the wave propagates. You can't stash it in a shoebox unless it's made of certain materials, and even then it keeps bouncing back and forth, never stopping. It eventually loses its energy and no longer exists.
When we get down to the quantum level we find that our particles are also behaving as if they were waves. But what does this mean? How can a particle "behave like a wave"? Does this make it a wave? The answer is complicated. Mostly because we don't live in the quantum world so the label "particle" and "wave" are two separate concepts that describe different things.
When we look closely we see that our "particles" behave, in certain circumstances, as if they were waves. They can exhibit interference and other wave-like phenomena. BUT, they also exhibit particle properties at the same time! They still have charge, spin, and if you stash them in a shoebox they only have a small chance of disappearing randomly. (Thanks, uncertainty principle)
So many times we say that they are both a particle and a wave because we are used to seeing those as two completely separate things. But as Phinds said they really aren't either one. They are quantum objects that exhibit certain behavior. It is only because we don't see the wave-like properties on our scale that we have the distinction between the two and have problems putting them together. If basketballs interfered with themselves while going through the basket in everyday life you would have no problem joining the two concepts together.