Negligible means which can be neglected or Ignored. Sometimes in physics, the force is so small that it's effect is of no consequence to the entire phenomenon. eg. when you throw an apple up, the apple attracts the earth but the earth does not move as the force applied by the apple is very small, or negligible.

What does it mean when a mass is negligible?

It really means photons, particles with no rest mass. Everything else might become questionable under some circumstances. E.g. neutrinos are very close to being massless, so they must move very close to geodesics, but in some rare cases pretty major deviations may happen.

Why can we neglect forces such as those holding a body together when we apply Newton's second law?

Why can we neglect forces such as those holding a body together when we apply Newton's second law of motion? Newton's 2nd law applies to external forces acting on a system. ... These two forces will "cancel" (the vector sum will be zero), so they will not cause acceleration (according to 1st law).