In a wireless communication system, the link margin, measured in dB, is the difference between the receiver's sensitivity (i.e., the received power at which the receiver will stop working) and the actual received power. A 15 dB link margin means that the system could tolerate an additional 15 dB of attenuation between the transmitter and the receiver, and it would still just barely work.
It is typical to design a system with at least a few dB of link margin, to allow for attenuation that is not modeled elsewhere. For example, a satellite communications system operating in the tens of gigahertz might require additional link margin (vs. the link budget assuming lossless propagation), in order to ensure that it still works with the extra losses due to rain fade or other external factors.
A system with a -negative link margin would mean the system is insufficient to transfer data, usually this means a better receiver is needed, with improved sensitivity.