Most generally, this means to "decrease the effect of", or "decrease the effectiveness of".

In biological neural terms, depression is the reduction in the ability of a signal (an action potential) at a synapse to affect the post-synaptic neuron. Depression is the reduction in the strength of a synaptic connection between the pre- and post-synaptic neurons connected through a synapse. Depression is the opposite of potentiation.

Depression is not inhibition. An inhibitory synapse can undergo post-tetanic, and long-term-depression (PTD, and LTD respectively). In other words, depression of an inhibitory synapse means to make it less likely to inhibit the firing of its neuron.