How is it possible to have negative mass for negative mass means something that has less mass even than nothing ? Does nothing has mass? And how is nothing a reference.
You take any mathematical model that includes mass as a parameter, e.g. Newton's Laws, Special Relativity, General Relativity, etc, and put in the mass as a negative number. The model will then make predictions about what will happen, but because you have chosen physically unrealistic starting conditions the predictions of the model will be physically unrealistic.
The point is that negative mass is not a description of anything that exists outside the fevered minds of unusually weird physicists.
All depend on the context. If Dirac's theory of antiparticles one has get a negative energy, so why one should not use a negative mass if it is convenient for the formalism. For example, in semiconductor physics, the negative effective mass is related to the curvature of the dispersion law describing dependence of the quasiparticle's energy on its momentum. Here, the anomalous dispersion corresponds to the negative effective mass.
Let's look at this from the non-relativistic perspective of Newtonian inertial mass.
Inertial mass, the quantity $m$ in the equation $F = ma$, is the amount of inertia where inertia is, loosely, the opposition to change in velocity.
The greater the magnitude of the mass, the less the acceleration for a given applied force.
For ordinary, positive inertial mass, the acceleration is in the same direction as the force. For a hypothetical negative inertial mass, the acceleration would be in the opposite direction of the force.
Interestingly, this would mean that the flow of energy is in the opposite "direction" than for the ordinary, positive mass case.
For example, increasing the speed of an ordinary positive mass requires work to be done on the object, increasing its kinetic energy. Increasing the speed of a negative inertial mass requires work by done by the object.
There's much more that can be said about negative mass in both the non-relativistic and relativistic contexts but, to address your statement:
for negative mass means something that has less mass even than nothing
Negative mass is counter-intuitive and perhaps unphysical but, mathematically, negative mass is something, not "less than nothing".