A "dead" ReLU always outputs the same value (zero as it happens, but that is not important) for any input. Probably this is arrived at by learning a large negative bias term for its weights.
In turn, that means that it takes no role in discriminating between inputs. For classification, you could visualise this as a decision plane outside of all possible input data.
Once a ReLU ends up in this state, it is unlikely to recover, because the function gradient at 0 is also 0, so gradient descent learning will not alter the weights. "Leaky" ReLUs with a small positive gradient for negative inputs (y=0.01x when x < 0 say) are one attempt to address this issue and give a chance to recover.
The sigmoid and tanh neurons can suffer from similar problems as their values saturate, but there is always at least a small gradient allowing them to recover in the long term.