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Generally, mass is the total energy of a particle or system in the frame of reference where its total momentum vanishes. A massless particle is one that can have arbitrarily small (positive) energy. For example, the energy of a photon is proportional to its frequency, which can be arbitrarily low. Note also that in General Relativity, the source of gravitation is not mass, but energy and momentum. In particular, light also has gravitational effects (although they are very small).

In the standard model, generally particle mass is caused by the interaction with the Higgs field; massless particles are those that don't interact with the Higgs field. There's one exception, though, and that is Neutrinos. Measurements show that neutrinos have mass, but according to the standard model they should not be able to interact with the Higgs field (the Higgs field couples left-handed and right-handed particles, and neutrinos only exist in a left-handed variant). As far as I know, the origin of Neutrino masses is not yet known.

I don't know enough about string theory to answer about the mass concept there, but is has to be some energy related to the vibration modes of the strings.