When error signals shrink to nothing or blow up to infinity as they travel back through many layers. Either way the network stops learning properly. This was the practical barrier that kept networks shallow for years, and solving it was what made deep learning possible.
Technical view
Repeated multiplication of Jacobians during backpropagation causes gradient magnitudes to decay or grow exponentially with depth. Mitigations include non-saturating activations such as ReLU, careful initialisation, normalisation layers, residual connections and gradient clipping. Recurrent networks suffered worst because the same weights are reapplied at every timestep, which is what gating mechanisms in LSTMs were designed to address.
Related
Part of AI Terminology Index, section: How the Machinery Works.