How can light be made up of particles? In order to explain photoelectric effect, Einstein suggested that light is made up of photons, but I don't understand how could this have made sense when you know that light is an electromagnetic disturbance. How can oscillating electromagnetic fields become particles?
 A: With the invention of lasers we no longer need the photoelectric  effect to have an easy experimental demonstration that light is composed of particles:
Have a look at this single photon at a time double slit experiment:


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

Note that on the right the usual diffractive pattern of light of this frequency appears, where there is an  accumulation of very many frames like the one on the left.
On the left the footprint of light comes quantized  and looks random, but as the number photons increases the diffraction pattern appears.
This is due to the quantum mechanical nature of light, and the classical electromagnetic wave can be mathematically shown to emerge from a very large number of photons.
In general, present day physics mathematical models can be shown to emerge from lower frameworks. For example Thermodynamics with its equations and laws, emerges from statistical mechanics. All frameworks at the bottom have quantum  mechanics as the underlying theory, and this can be proven mathematically. For example see how the classical field emerges from the quantum mechanical in this blog entry.
A: I tend to think particle-wave duality this way:
The actually physical state of the system is some abstract mathematical object. In old fashion quantum mechanics, it's a wave function (or ray or whatever). In quantum field theory, it's the quantum field (super position of many classical field).
What do we mean by a free particle? One way to informally "define" it, is to assume a free particle need to travel without chancing its profile (profile of whatever the abstract object that describes the physical state) that much.
An intuition in quantum mechanics is "superposition of very similar energy eigenstate can travels a long distance without chancing its profile". If you have learned about wave packet and group velocity in the context of partial differential equation of wave, this intuition should be intuitive to you. If not, you can believe me for now. If you don't know what energystate is, then you just need to know it's something that can be calculated from the fundamental law of physics in quantum mechanics.
Therefore, in a quantum theory, free particle manifest itself as energy eigenstate. Again, superposition of "similar" eigenstate would create something that travels without changing its other properties (a particle).
In old fashion quantum mechanics, the wave like behavior comes from wave function, the particle like behavior comes from wavepacket (the super position mentioned). Also wave propagation through spatially dependent potential in schrodinger equation, can be transformed to superposition of particles scattering with the potential. (this is a change of basis from position basis to momentum basis (energy eigenstate basis) )
In quantum field theory, the wave like behavior comes from the quantum field itself. The particles come from solving its energy state and the superposition mentioned. A wave like quantum field (a superposition of classical field) with interaction can be transformed into superposition of particle state scattering with each other (like in the typical fock space notation) through a change of basis from (field basis to particle basis (energy eigenstate basis))
A: Of course annav's answer is right, but let me add a few things:
Photons are as per the Standard Model, elementary particles, point like, massless, with no spatial extent or substructure.


The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force (even when static via virtual particles). The invariant mass of the photon is zero; it always moves at the speed of light within a vacuum.


What you are asking about is wave-particle duality. Photons can take characteristics of waves and particles too.


Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of both waves and particles. For example, a single photon may be refracted by a lens and exhibit wave interference with itself, and it can behave as a particle with definite and finite measurable position or momentum, though not both at the same time as per Heisenberg's uncertainty principle. The photon's wave and quantum qualities are two observable aspects of a single phenomenon—they cannot be described by any mechanical model;[2] a representation of this dual property of light that assumes certain points on the wavefront to be the seat of the energy is not possible. The quanta in a light wave are not spatially localized.


https://en.wikipedia.org/wiki/Photon
There are experiments where we can see the wave characteristics and there are experiments where we can see the particle characteristics. In reality photons have both.


Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be partly described in terms not only of particles, but also of waves.


https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality
There are experiments, that will show wave characteristics for light and some that will show particle (but in reality they show both, we just try to observe one of them more), for example there is the black -body radiation problem that could not be solved without the quantum nature of photons, and the photoelectric effect.
A: It's up to you how you think about electromagnetic radiation. The more EM phenomena you observe, the more you will relate those phenomena to light as particles or aa waves.

In order to explain photoelectric effect, Einstein suggested that light is made up of photons

The photoelectric effect is one of these phenomena which where not explainable with light as a wave:


In 1902, Lenard observed that the energy of individual emitted electrons increased with the frequency (which is related to the color) of the light.
    This appeared to be at odds with Maxwell's wave theory of light, which predicted that the electron energy would be proportional to the intensity of the radiation.
    In 1905, Albert Einstein solved this apparent paradox by describing light as composed of discrete quanta, now called photons, rather than continuous waves. (Wikipedia)


The laser radiation is another phenomena for the quanta nature of light. Excited electrons fall back and emit photons. The emission of photons from electrons and other subatomic particles is the only reason for all the electromagnetic radiation around us. Broadcasting, WiFi as well as light from a bulb is based on the acceleration of electrons which emit photons.

... but I don't understand how could this have made sense when you know that light is an electromagnetic disturbance. How can oscillating electromagnetic fields become particles?

Light as a disturbance of an overall existing EM field comes into play with QM:


Quantum mechanics was initially developed to provide a better explanation and description of the atom, especially the differences in the spectra of light emitted by different isotopes of the same chemical element, as well as subatomic particles. In short, the quantum-mechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter. (Wikipedia)


However, for phenomena around the nucleus the declaration of a in general existing EM field is connected with the subatomic particles, which emit and absorb photons. To generalize this approach to an empty space, declaring an EM field without charges and masses, seems somewhat risky. At last it confuses.
What you have to beginn with is to realize that


*

*EM radiation is emitted in packages, called photons

*each photon has an electric and a magnetic field component which oscillates (the intensity of these field components changes from maximum to zero to the opposite maximum and so on), the plotted values of these intensities are sine waves

*in the double slit experiment the observed light on the screen is made of photons (see Anna’s answer), from the intensity distribution pattern on the screen it is deduced about the wave nature during the interaction with egdes (see Thomas Young).

