BSD stands for Berkeley Software Distribution, and it was a UNIX software compilation released by the Computer Software Research Group at the University of California, Berkeley between 1977 and 1995. The first one was 1BSD (First Berkeley Software Distribution) compiled by Bill Joy, but only as an add-on to Sixth Edition Unix from Bell Labs. In 2BSD Bill Joy added a C Shell and the iconic vi text editor.
As BSD evolved it became a complete UNIX operating system in its own right. The last release from Berkeley was 4.4BSD-Lite Release 2, and it contained no proprietary code from AT&T, and was freely available under the permissive BSD license. Since then other projects, descendants of the original BSD, continued the development and they are what BSD today generally refers to. Most popular of these are FreeBSD, NetBSD, and OpenBSD.
BSD was the first to include support for the Internet Protocol stack in form of its Berkeley sockets, which made it easy to read and write files over the network. It is also easier to natively run software from other operating systems on BSD thanks to its binary compatibility layer. The very permissive nature of the BSD open source license enabled widespread use of its code in various other software projects. Apple's OSX and iOS, for example, are based on BSD code.
Originally based on top of 386BSD, and since version 2.0 on 4.4BSD-Lite FreeBSD exists with a goal of providing a complete operating system that can serve any purpose without any strings attached, free in every sense of the word. FreeBSD originated the ports collection system for easy download, building, and installation of software packages that continues to be one of the easiest and most sophisticated ways to install software in the UNIX world. This was adopted by NetBSD and OpenBSD as well.
FreeBSD also uses a rather open model of development by letting hundreds of "committers" make changes to FreeBSD at any time as needed. The selection of committers and resolution of any disputes is managed by the elected Core Team.
FreeBSD is the most popular BSD version used, and also served as the basis of many other operating systems such as most notably the Apple's OSX. There is also a number of FreeBSD variants such as the desktop oriented PC-BSD created to be easy for everyone to use.
NetBSD was founded after FreeBSD with major emphasis on portability at a time when FreeBSD was mostly focusing on the x86 architecture. NetBSD runs on so many platforms that they have a slogan saying "Of course it runs BSD". This makes it particularly suitable for computer research because it readily runs on both old and new architectures alike. NetBSD also uses the pkgsrc package management system originally based on the Ports collection system from FreeBSD.
In 1995 Theo De Radt forked the NetBSD project to create OpenBSD. Particular focus was put on security as well as strong emphasis on great documentation, code correctness, and open source licensing. The project has also spawned a number of key widely used security tools like OpenSSH, OpenNTPD, PF, and most recently the LibreSSL fork of OpenSSL after the Heartbleed bug fiasco. Unlike FreeBSD the project is more strictly managed by Theo De Radt himself. OpenBSD also supports about 20 different hardware architectures.