Raw and Pure magic are two names for the same thing: magic in its natural state, as it exists when a mage first calls it into being. "Raw" carries the connotation of more reckless use, whereas "Pure" suggests that the user has a high level of control over his power.
Pure magic essentially does whatever the user wishes. It can heal wounds, slash foes from a distance, or blast through walls. Most complex magic utilizes pure magic rather than an elemental form.
Any magic that does not produce a noticeable visual effect also usually uses pure magic. For example, telekinesis, or techniques that rely on deconstructing or reforming an object on a molecular level, typically use pure magic.