What is radiationless transition in Laser? I am studying Lasers and I am very confused regarding radiationless transition. Why electron does not emit photon while downward transition from higher state to metastable state and in which form the energy is released ?
 A: If I'm not mistaken and we talk solid-state lasers you really mean non-radiative transitions. 
In this case excess energy is dissipated in various manners, but the simplest is phonon (or multi-phonon) relaxation. This means energy transfer to the crystal lattice. This causes heating of the laser medium.
This is usually adverse effect, but in some cases it helps with obtaining population inversion. This may mean populating upper laser level from higher lying level (for example samarium ions), or depopulating lower laser level.
There are also possible processes called cross-relaxation in which energy from one ion in excited state is transferred to another. This mechanism strongly depends on ions in question but can be both adverse and advantageous. In this case the energy just migrates to neighbors, and if the energy matching between the levels is not perfect phonons are generated. 
This can cause ion in upper laser state to first undergo cross-relaxation leading to two ions in excited stated of about half the initial energy, which then decay by phonons, causing heating of the medium. In thulium doped  materials however this can lead to efficient pumping of upper laser level which is final state of such a cross-relaxation process. 
And even concentration quenching can be useful. For example erbium lasers operating at 2.9 microns have lower laser level which is first excited state. It can be used to achieve laser action around 1.53 microns itself, so it's prone to population inversion. This makes pumping the upper laser level (that is second excited state) in this scheme to population inversion much tougher. If the doping is high enough however, the energy is being removed from lower state by transfer to neighboring ions allowing the laser to work in continuous mode.
A: This is a so-called Auger transition. The first electron making a transition from a higher state into the vacant hole releases energy $\Delta E$. This energy is transmitted - via a virtual photon - to another electron, which then makes a transition into the continuum (under the condition that the binding energy of this second electron is less than $\Delta E$). Hence, no photon is emitted by the atom, the energy is released in the form of the binding energy of the second electron plus its kinetic energy in the continuum.
