XBMC is the old name for what is now known as Kodi. XBMC is still the proper name if you are referring to an old version of the program (e.g., XBMC v13), but versions as of v14 and the project itself is known as Kodi.
Kodi® media center, formerly known as XBMC Media Center, is an award-winning free and open source cross-platform software media player and entertainment hub for digital media for HTPCs (Home theater PCs). It uses a 10-foot user interface designed to be a media player for the living-room, using a remote control as the primary input device. Its graphical user interface (GUI) allows the user to easily browse and view videos, photos, podcasts, and music from a harddrive, optical disc, local network, and the internet using only a few buttons. The Kodi project is managed by the non-profit XBMC Foundation, and developed by volunteers located around the world. Since its creation in 2003 more than 500 software developers have contributed to Kodi, and around 60 developers on regular basis. Also 200-plus translators have worked to expand its reach, making it available in more than 72 languages.