Energy and mass are interrelated. As everything has energy could any object be massless? For example a photon is a packet of energy but still it is considered to be a massless particle. Why is it so?
Answer
There is only one mass. Lets make this clear. The concept of "relativistic mass" is not really a useful concept in my opinion. The invariant mass, or simply the mass, is defined as (in natural units, so $c = 1$):
$$E^2 - p^2 = m^2$$
The reason this is a much more useful definition for a mass, is because this quantity is Lorentz invariant, meaning it has the same value in every reference frame. If you define mass in any other way you are going to run into unnecessary trouble.
For the photon, this invariant mass is assumed to be 0, so its energy, $E$, gets a contribution only from the momentum of the photon, hence $E = p$. There are justifications for why we assume the photon has zero mass. The photon only has 2 degrees of freedom; the longitudinal polarisation does not exist precisely because the photon is massless. We also have other reasons to believe the photon is massless. Some laws of electromagnetism would have to be modified as well if the photon isn't massless, an example of which would be Coulomb's law. Hence Coulomb's law provides a good test of the photon mass (refer to this paper) which has been assigned the upper limit of $m ≲ 10^{−14}$ eV/$c^2$.
For other particles this is not the case; since they also possess this intrinsic mass they get contributions to the energy from that quantity as well and therefore $E^2 = p^2 + m^2$.
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