How is light emitted by an atom?

   Atoms emit radiations when they are excited. An atom can be excited in different ways - Exothermic chemical reactions, collision of energetic particles with atoms, collision between atoms, by energetic incident photons etc. An atom has enormous number of resonant columns in it and each of them has their own unique natural frequency. We can call a resonant column in an atom as a shell. If a shell has electron(s) it is called electron shell and if it has not is called transitory shell. When an atom is excited its shells start to oscillate. Since each of the shell has their own unique natural frequency, when excited a shell oscillates in its natural frequency. If a shell has one or more electrons in it, the excitation of that shell also causes the electron(s) in it to oscillate with the shell which causes the emission of light.

How do electrons emit light?
   Because of the positive charge of the nucleus, the electric field of an electron will be directed to the nucleus and magnetic field will be horizontal to the nucleus. Since the electric field is directed to the nucleus, when oscillates the electric field of the electron will be parallel to the oscillation and the magnetic field will be perpendicular to the oscillation. This oscillation of the electron creates a transversely-oscillating magnetic line and the oscillating magnetic line (OML) is radiated to space. The OML is created at right angle to the oscillating electron. i.e. there is a 90 degree angle between the oscillation of electron and the emission of photon.
   Since there is density differences between inner and outer regions (see structure of atom), when an electron is excited in an atom, for every oscillation towards the direction of the nucleus, the high-density space matter in the inner region of the atom expels the electron to an outer low-density space matter region. 
   When the electron shell of an atom is excited, the electron in the shell emits a photon in the natural frequency of that shell and jumps to an outer transitory shell. If that transitory shell is simultaneously excited by some ways, the electron again emits a photon in the natural frequency of that transitory shell and jumps to a further outer transitory shell. In this way, an atom with only a few electrons can emit a wide range of spectrum lines when continually excited (see line spectrum of hydrogen atom). Since the electron jumps to an outer region with the emission of a photon, the electron loses its excitation simultaneously. This is the reason for the photon/quantum nature of radiation emitted by an atom.

See also: What is radio wave?