Vanishing gradient
Also: vanishing gradients, exploding gradients, vanishing gradient problem
Backpropagation multiplies one factor per layer or time step; if those factors are below 1 the product decays exponentially (vanishing), and if above 1 it blows up (exploding). ReLU, residual connections, LSTMs and careful initialization counter it.
Example
The sigmoid's derivative is at most 0.25, so through 10 sigmoid layers the gradient can shrink by 0.25¹⁰ ≈ 0.000001: the first layer gets a millionth of the signal.