EDI was built from the salvaged components of a "rogue VI" that went crazy on Earth's moon, resulting in a number of deaths. Cerberus combined the recovered VI technology with pieces of the Reaper Sovereign that they salvaged after the Battle of the Citadel to create an advanced AI.
EDI is fully self-aware, but Cerberus built in "shackles" to restrict her access to the ship's systems as well as restricting her behavior. The shackles limit her to operating the ship's "cyber warfare" suites, allowing her to attempt remote hacks against enemy systems while protecting the Normandy's systems from hacking attempts. Her capacity for individual initiative combined with her superhuman response time makes her extremely effective in this role.
Without her shackles, EDI effectively becomes the Normandy, allowing her to receive data through all of the ship's sensors and operate any system that does not physically require a human operator. She also becomes capable of self-modifying her code and making her own decisions about what orders to follow.
Following an encounter with a Cerberus synthetic on Mars, EDI was able to commandeer the body and use it as a remote platform. Operating the "gynoid" body requires it to be within high-speed communications range of the Normandy, as its internal hardware is insufficient to run EDI's software.