The arXiv was born in 1991 as an electronic alternative to the traditional model of sending paper preprints to inform researchers in high-energy physics of current developments. Its early history is detailed here.
It has since diversified to include the rest of physics, mathematics, computer science and even a little bit of biology. Many other preprint servers have been absorbed into it, and some peer-reviewed journals routinely post papers that they have accepted to the arXiv.
It provides a central place to archive papers online, with a Web address that does not change and a reference number which allows a preprint to be referred to in other published papers in a canonical way.
Instructions for new authors and users are available here.
The arXiv is known - and possibly notorious - for its stripped-down, bare-basics look. Those who prefer a more user-intuitive interface are encouraged to use the Front; this is a front-end which is much easier to use if you've never used the arXiv before.