How does the sensitive plant detect vibrations?
In fact, the idea of a plant nervous system is quite serious and constantly developing; of course those are rather local, simple signal pathways rather than an "animalian" centralized global network, but they use similar mechanisms -- depolarisation waves, neurotransmitter-like compounds, specialized cells... Here is a review paper by Brenner et al.

In the case of Mimosa, there is a good paper summing up Takao Sibaoka's long research of the topic.  

In short, it seems that its petioles' phloem has cells which have polarized membranes and can trigger depolarization due to a mechanical stimulation. The signal then propagates to the corresponding pulvinus by a mixture of electrical and Cl- depolarization waves.

In the pulvinus, this signal triggers a second depolarization which coordinates the pulvinus' cells to trigger water pumping responsible for the leaf drop. 

The transmission to the adjacent leaves is most likely mechanical, i.e. the movement of one dropping leaf excites another.

References:


 Brenner ED, Stahlberg R, Mancuso S, Vivanco J, Baluska F, Van Volkenburgh E. 2006. Plant neurobiology: an integrated view of plant signaling. Trends in plant science 11: 413–9.
 Sibaoka T. 1991. Rapid plant movements triggered by action potentials. The Botanical Magazine Tokyo 104: 73–95.