In startup culture, it usually refers to a group of individuals who underwent a formative experience in one company, then went on to start separate individual companies where they all cross-invest, cross-advise, and generally help out each others' companies. Term was originally used to refer to the PayPal mafia [1, notable members include Peter Thiel, Max Levchin, Elon Musk, Chad Hurley, Reid Hoffman, Jeremy Stoppelman, Yishan Wong, notable descendants include SpaceX, Tesla, YouTube, Yelp, LinkedIn, and arguably Facebook]. Since expanded to the justin.tv mafia [2, members = Justin Kan, Emmett Shear, Kyle Vogt, and Michael Seibel, descendants include Twitch & Cruise]. Arguably the Fairchildren (descendants of Fairchild Semiconductors - Intel, AMD, National Semiconductor Kleiner-Perkins, Sequoia Capital, and by extension Apple, Google, Cisco, Netscape, etc.) and the descendants of General Magic (E-bay, Android, iPod/iPhone, Nest, WebTV, and the United States Digital Service) could also be termed "mafias", although the aren't usually referred to as such.
It's not just starting a second company - it's that a group of people who were all bound together by one company end up starting second companies, and they continue to go on to help each other and collaborate in their later ventures.
[1] https://en.wikipedia.org/wiki/PayPal_Mafia