I am able to understand how light can be modeled to have wave characteristics from Young's double slit experiment.
But I am unable to comprehend how we can understand light to have particle characteristics from the photoelectric experiment. How is it wave character not able to explain the phenomena observed in this experiment? And how is it that particle nature defeats the wave theory?
There have been attempts to describe the photoelectric effect by taking the EM field as a classical wave. For a discussion see a previous question "Can the photoelectric effect be explained without photons?". One of the answers describes that the photoelectric effect can be well explained considering the EM field more or less as a classical wave. To explain other experimental data though a quantized version of EM waves is needed.
On the second part of your question "And how is it that particle nature defeats the wave theory?" The above does not mean that these "wave quanta" (or photons) are particles in the sense of being localized objects flying around in space until they "hit" an atom kicking out an electron. A photon as a quantum of the waves is not localized or trackable as what one would think of a particle. Some physicists refere to these photons as particles which could lead to confusion (e.g. through which slit did they fly in these double slit experiments), but the bottom line is that even if you call these quanta particles they certainly do not defeat the quantum version of the wave theory.
Why don't you look a this video an A levels tutorial for the photoelectric effect.
Summary of the observations of the puzzling photoelectric effect:
The electrons were emitted immediately - no time lag!
Increasing the intensity of the light increased the number of photoelectrons, but not their maximum kinetic energy!
Red light will not cause the ejection of electrons, no matter what the intensity!
A weak violet light will eject only a few electrons, but their maximum kinetic energies are greater than those for intense light of longer wavelengths!
The basic point is that the effect disappears at a threshold frequency and is not dependent on the intensity of the light impinging on the metal.
A wave formalism cannot explain all of these, because the energy in waves is additive, the more intense the beam, the more energy it delivers, but the photoelectric experiments show that this is not true for "light wave" + "electrons in metal" scattering.
No matter how strong the incoming beam of light, if it is below a threshold in frequency (depending on the metal), the electrons will not budge.It shows a one to one correlation that only a particle model can explain.
Take it by the other way, for the emission as well for the absorption of EM radiation photons are a good description. The photoelectric phenomenon is a good example for the point that EM radiation is made of photons.
The description of EM radiation as a wave has some weaknesses. For a thermic source of EM radiation one will not be able to measure nor the amplitude nor the wavelength directly. For a monocromatic source (with small aperture) one can measure behind a double slit the distances between the fringes of the intensity distribution which is explained by the interference of outgoing from the two slits circular waves (Huygens principle). But since the phenomenon of fringes appears behind every single sharp edge and even for a stream of single emitted photons, the explanation with Huygens principle is not holdable for this case. Than more the evidence for wave nature is made from a pattern which intensity distribution has the equation of a wave (a sin equation).
But really there is EM radiation which directly measurable frequency. Radio waves are produced by accelerating electrons in an antenna rod by a wave generator fore and back. This induces the emission of photons. Such a EM radiation clearly has the properties of energy transfer with a wave characteristics.
The description of monochromatic EM radiation with a associated wavelength /frequency is helpful but not necessary. The description by the energy of the involved photons is enough.
More about photons, EM radiation and radio waves see this answer.
