FLV stands for FLash Video and is a proprietary streaming video format from Adobe. This format was used to store and share video clips on sites like YouTube, Google Video, Yahoo! Video, and many other sites. This changed with the advent of HTML5 and MP4 technologies.
Normally you cannot download the actual FLVs, but there are applications and hacks that let you do so. There are also several FLV players and converters (that convert FLVs to other movie/video formats) available.