What is a preprint, and why should I publish one?

According to Wikipedia, a preprint is

[...] a version of a scholarly or scientific paper that precedes publication in a peer-reviewed scholarly or scientific journal. The preprint may be available, often as a non-typeset version available free, before and/or after a paper is published in a journal.

A preprint makes a manuscript widely available independently of its formal publication. This provides greater and earlier exposure compared to the traditional route, which has advantages and downsides.