The answer by Alfred Centauri is correct, it is how physicists now view the use of the term "mass", but it needs some clarification on the usage.
- $$\vec p = \frac{m\vec v}{\sqrt{1 - \frac{v^2}{c^2}}}$$
$$\vec p = \frac{m\vec v}{\sqrt{1 - \frac{v^2}{c^2}}} \tag{1}$$
When special relativity was first studied the equation
- $$E = \frac{m{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}}$$
$$E = \frac{m{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}} \tag{2}$$
(where m is the rest mass the measure of the four dimensional vector describing the particle/system in motion), gave a new definition for the energy of moving objects.
From Newtonian physics we know that p=m*v
$\vec{p} = m \vec{v}$, so it was "natural" to think of the value multiplying the velocity vector in momentum as a mass: they called it "relativistic mass" because as far as classical inertial systems go the particle will behave to changes in motion as if its mass is increasing according to formula 2): more and more energy has to be supplied to reach the value of v. It is the relativistic mass that increases. In the rest mass of the particle, its mass is always m.
The use of the term "relativistic mass" is being dropped, because it gives rise to confusions as in your question. Each particle has a mass in its center of mass system that does not change .
A better term might be "apparent mass", how an observer would see a particle approaching the speed of light : as if it had more and more difficulty reaching there.
By the way, in the lab we observe relativistic elementary particles which are point particles, like the electrons.
The particle accelerator known as the Large Electron Positron (LEP) collider at the Centre Européenne pour la Recherche Nucléaire (CERN) laboratory near Geneva could propel electrons to 99.999999999 percent of the speed of light.
Their relativistic mass increases, but they are still measured as point particles with rest mass m.
