A forbidden line (or nebular line) is a spectral line appearing in astronomical bodies that are not reproduced in the lab because labs cannot sufficiently rarefy a sufficient amount of gas to produce observable lines. Mechanisms producing them are termed a forbidden mechanisms. The lines form from electron orbit transitions that are sufficiently improbable that their timescale is much longer than that of collisions between atoms that trigger more probable transitions. Thus the phrase "highly improbable line" would be more descriptive than forbidden line. They occur when an electron's state of excitation is metastable, i.e., energy is required to "get over a hump" to make the move to a lower-energy state that would produce the line, and outside influences such as a collision is likely to trigger some larger transition.
In space, a gas cloud can be large as well as thin enough that collisions are significantly less frequent, allowing sufficiently many of these improbable forbidden transitions that we can detect them.
They are seen in the thin upper Earth atmosphere as well as HII regions and planetary nebulae.
The square-bracket form [N II] indicates forbidden transitions by the given species (e.g., "N II" being nitrogen with a state of ionization of 1, i.e., singly ionized).