The most well-known preprint server is probably arXiv (pronounced like ‘archive’). It started as a server for preprints in physics and has since expanded out to various subjects, including mathematics, computer science, and economics. The arXiv server is now run by the Cornell University Library and contains 1.37 million preprints so far.

The Open Science Framework provides an open-source framework to help researchers and institutions set up their own preprint servers. One such example is SocArXiv for the Social Sciences. On their website, you can browse more than 2 million preprints, including preprints on arXiv, and many of them have their own preprint digital object identifier (DOI). In cases where the preprint has now been published it also links to the publication’s DOI.
Cold Spring Harbor Laboratory set up bioaRxiv, a preprint server for Biology in 2013 to complement arXiv. The bioaRxiv server has a direct transfer service to several journals such as Science and PNAS and a bit over 60% of papers in bioaRxiv end up published in peer-reviewed journals.
In more recent years a lot of new servers have popped up covering almost every field including the social sciences, arts, and humanities fields. Here’s a quick overview of some of the rest:
arXiv -> Mathematics, Computer science, and economics, Physics
EngrXiv - Engineering
ChemRxiv - Chemical sciences
PsyArXiv - Psychological sciences
SportaRxiv - Sport and exercise science
PaleoarXiv - Paleontology
LawArXiv - Law
AgriXiv - Agricultural sciences
NutriXiv - Nutritional sciences
MarXiv - Ocean and marine-climate sciences
EarthArXiv - Earth sciences
Preprints.org - Arts & Humanities, Behavioral Sciences, Biology, Chemistry, Earth Sciences, Engineering, Life Sciences, Materials Science, Mathematics & Computer Science, Medicine &, Pharmacology, Physical Sciences, Social Sciences