Ultraviolet divergence
An ultraviolet (UV) divergence arises when a loop integral receives unbounded contributions from arbitrarily high momenta — equivalently, arbitrarily short distances. The integrand doesn't fall off fast enough at large loop momentum, and the integral diverges.
Physical meaning
UV divergences signal that we're extrapolating the theory beyond its domain of validity. QED is an effective field theory — it breaks down at very high energies where new physics (electroweak unification, ultimately quantum gravity) takes over. The divergences are absorbed into a finite number of physical parameters (mass, charge, field normalization) through renormalization. In dimensional regularization they appear as $1/\epsilon$ poles where $d = 4 - \epsilon$.
Resolution
Renormalization. The bare parameters in the Lagrangian are redefined to absorb the divergences, leaving finite predictions for physical observables. QED is renormalizable: only three parameters (electron mass, electron charge, field normalization) need renormalization, and this suffices to all orders.