The use of preprint servers is one of the main phenomena that have been observed for the last two years in the academic publishing world. Nevertheless, the concept is not new: The first and still the most famous preprint server, ArXiv, was created in 1990 by Los Alamos scientists, and despite its 90' old fashioned design, remains the most used of all preprint servers.

Today, almost every field of research has its own preprint repository : BioRxiv for biology, SocArxiv for Sociology, ChemRxiv for chemistry, etc. HAL can also be used as a preprint server. These servers play a major role in the dissemination of scientific results given that the average peer review process can take a long time (up to one year for some journals).

If you deposit your paper before submission on one of these servers (you can do it at any time and without embargo), your idea will be time-stamped and your paternity will be testified (have no fear of being "scooped"). Besides several of  these servers will be able to attribute a DOI to your work that will make it more citable. Nowadays, preprints are valuable assets to be cited when you apply for a job or a grant. (See this post from the publishing blog of the Royal Society.)

On the other hand, only few publishers will refuse a manuscript upon the pretext that it has been already deposited elsewhere in open access. On the contrary, one can observe on sites like SherpaRomeo a shift ot the major part of them towards a more liberal policy about preprints.

But the main advantage would be to stimulate a conversational pre-peer review process with the researchers who work on the same topic. This input can make your work better and more likely to be accepted by a publisher.

Moreover some publishers browse preprint servers in order to find relevant and accurate papers for their journals. If their journals are only made of papers extracted from preprint servers, they are called "epi-journals"

For further reading, see the Ten simple rules for considering preprints published on PeerJ