Infrared divergence
An infrared (IR) divergence arises when loop integrals receive unbounded contributions from arbitrarily low momenta — arbitrarily long distances. In QED this happens because the photon is massless: virtual photons with nearly zero energy cost nearly zero action.
Physical meaning
IR divergences are not a sign of missing physics — they're a sign you're computing the wrong observable. The amplitude for a process with exactly zero soft photons emitted is zero: any scattering of charged particles necessarily produces a cloud of infinitely many soft photons, each carrying infinitesimal energy. The IR divergence is telling you that this process never happens in isolation.
Resolution
The Kinoshita-Lee-Nauenberg (KLN) theorem: IR divergences cancel exactly when you sum over all degenerate final states — processes that are experimentally indistinguishable because your detector has finite energy resolution. Virtual soft photon loops cancel against real soft photon emission. The physical cross section, summed over soft photon multiplicity, is finite.