A" The fact that particles (and waves) can appear like each other is one of the enduring philosophical (and practical/scientific) mysteries of quantum mechanics, with a number of different interpretations.
Firstly, it has to be said that there is no denying that this is in some way a real thing: you can take something that is ostensibly a particle (like an electron) and you can perform various experiments on it. For example, if you collide it with another electron, they act in much the same way as colliding billiard balls, i.e. as particles. But, if you pass an electron through a pair of slits, it interferes with itself just like a wave i.e. like light, sound and water waves. This is a slight simplification - if one electron passes through the slits, and hits a screen, you will find it at almost any location on the screen - even at angles that would be impossible if it was just a particle moving in a straight line through one or the other slit - but if you repeat this experiment over and over you will see that the distribution of impact points is just like that for interfering light - it is as if the electrons are waves and interfere with themselves in the same way - even if the electrons arrive so rarely that there are never two of them in the air at once!
Conversely, light usually acts as a wave, but in some situations it acts like a particle (photons).
My favored interpretation of this affect is one of the more common ones. In quantum mechanics, a particle's position (and momentum) are not necessarily certain values - instead, they are represented by a probability distribution that describes how likely a given result is when the position (or momentum) is measured. As time progresses, this probability distribution moves like a wave (as described by the Schroedinger Equation), hence the wavelike interference. In the case of the double slit experiment, as we dont know which slit the electron went through, we are obliged to calculate the probability of it arriving at a given point via either path - leading to maxima and minima in the likelyhood of it hitting the screen in different places.
If we knew which slit the electron passed through, then we would only have to make the calculationfor one path, mathematically removing the interference. Amazingly, this is exactly what is seen in experiment. How it is that our knowledge can influence events like this is still under intense debate, but it certainly does happen!