In an approximate way, light is both a particle and a wave. But in an exact representation, light is neither a particle nor a wave, but is something more complex. As a metaphor, consider a cylindrical can of beans. If you hold the can sideways, force a friend to only look at its shadow, and ask him what shape the object has, he will respond “rectangular”. But now turn the can ninety degrees, have a second friend look at just the shadow, and he will tell you the can is “circular”. Now have your two friends debate each other about the true shape and they will not make much progress. Which one is right? They are both right in away and both wrong in a way. A cylinder is circular as seen from one angle, and rectangular from another angle, but in reality it is much more than a circle plus a rectangle. It is something more complex: a three-dimensional shape that can’t be fully described through two-dimensional shapes such as circles and rectangles. The problem is that your friends were looking at shadows of the can of beans and not the object itself. A shadow is a two-dimensional, collapsed representation of a three-dimensional object. The case is very similar when it comes to quantum particles such as light. To say light is a particle is to look at it as a collapsed representation of a more complex entity. Similarly, to picture light as a wave is to treat it as a simpler object than it really is.
Light sometimes acts like a wave and sometimes acts like a particle, depending on the situation. This only makes sense if you accept that light is something more complex; something that from a certain perspective looks wave-like and from another perspective looks particle-like. So what really is light? The question is hard to answer without delving into complex mathematics. The situation is much like the proverbial elephant and the blind men. One blind man feels only an elephant’s leg and declares the elephant to be a tree. Another blind man feels the elephant’s tail and declares it to be a rope. Still another feels a tusk and declares the animal to be a spear. All the blind men are partly right and also partly wrong because they don’t have complete information. But how do you explain an elephant to a blind man without him climbing all over it and feeling every inch for himself? This is the difficulty physicists find themselves in explaining quantum particles to people unable to solve the mathematics for themselves.
Light is a complex-valued probability distribution that has quantized (discrete) properties such as energy. The smallest piece of light is called a photon. Like a wave, a photon experiences diffraction (bending around corners), interference (fringed patterns), refraction (bending when entering a material), reflection, dispersion (wave-shape spreading), coherence (lining up of phases), and has a frequency. Like a particle, a photon contains a fixed energy, a fixed momentum, a fixed spin, and can be measured to have a single fixed location in space. The wave-like and particle-like traits of a photon trade off according to the Heisenberg Uncertainty Principle. This means that the more you force a photon to act like a particle, for instance by confining it in a small box thereby lowering the uncertainty in its position, the less it acts like a wave.
In the famous Young’s double-slit experiment, a coherent beam of light is directed through two slits and then onto a photographic plate. When each photon hits the plate, it makes a single, point-like mark, indicating that the photon interacted with the plate as a particle. But the overall pattern of marks on the plate is that of an interference pattern, which is only possible if the light is a wave. The interference is the result of two beams being created by the two slits, which spread out from the slits and interfere with each other. Even more remarkably, if we dim the light until we are only sending through one photon at a time, we still get an interference pattern. This means that a single photon goes through both slits at the same time, interferes with itself in a wave-like way upon emerging from the slits, and then makes a single mark on the plate in a particle-like way. If this sounds nonsensical to you, it is because you are still picturing the photon as just a particle or a wave. Because the photon is a fluctuating probability distribution with quantized properties, it can do all these things in a completely sensible way.
Amazingly, all quantum objects from electrons to protons behave as quantized probability distributions, and not just photons. If you find a quantum particle/wave hard to visualize, don’t let this difficulty tempt you to dismiss quantum theory as nonsense. Quantum theory has been experimentally verified in hundreds of laboratories for almost a century now. Additionally, the semiconductor chip inside the computer you have in front of you right now crucially depends on quantum theory being right. To dismiss quantum theory as quackery because it’s concepts are hard to visualize is to say that computers don’t exist.