A Focus (plural Foci) is a magical tool used by a mage to focus his paradigm to achieve a desired effect via Awakened Magic. Foci are thus items, rituals, and practices that mages use to tap their magical power.
A focus may be a totally mundane object or ritual, with no power of its own, but because the mage believes not only in the focus' power but in her ability to use it, the focus opens the door to magic.
However, Foci are not necessary for casting magick, they only make it easier. Mages can transcend their need for a Focus for a short time to express their own Will, but the belief in her own need makes it more difficult for him to do magic unaided.
Technomancers use Devices as their foci and are much more bound to their tools. Because of the nature of their paradigm, all Enlightened Science ("magic") requires secondary tools. Not until a Technomancer reaches the level of Archmage can they overcome their need for Devices.
Archmages and even some Masters have demonstrated the ability to work magick without any help, channeling their own Awakened Will instead.