Red Magic as a skillset tends to appear in two quantities: a "White and Black Magic" (白黒魔, しろくろま, Shiro Kuro Ma?) skillset (called B/W Magic in the Bravely series) that only has access to lower level spells but can use both in one ability slot, and as "Red Magic" (赤魔法, あかまほう, Aka Mahō?) which is used in games where Red Mages use a specific set of skills (and often features spells not available to either White or Black Mages).
Games with Red Mages but no Red Magic skillset tend to use a single Magic command that works in a similar manner to the "White and Black Magic" skillset except the levels of magic available are defined by the job rather than the command.
Final Fantasy XI has a single Magic command while the available spells are determined by the job. Each spell is classified as Black Magic or White Magic, however not all of these spells are available to the Black Mage and White Mage respectively, and the Red Mage has exclusive access to some of them, such as the Gravity spells (black) and the Gain spells (white). These exclusive spells are colloquially referred to as "Red Magic".
Red Magic is the specialty of the Red Mage class. It enables the user to cast both Black and White Magic, though only up to level 3, therefore limited to half of both types of magic commands. Upon mastering the job the Red Mage learns an upgrade to Red Magic, Dualcast, to cast two spells in one turn.
Red Mage's action ability is Red Magic, which is mostly comprised of weak elemental spells like Fire. Due to its useful Doublecast ability, Red Magic is an ideal secondary action ability for magic-based viera jobs, such as the Summoner.
Red Magick returns as the innate skillset of the viera Red Mage. The skillset meets some changes, such as the substitution of the Sleep and Poison debuffs for Silence, and the substitution of Barrier for the Shell and Protect spells.
Red Mages can cast White and Black magic spells up to Level 4, although further levels exist unused in the game code.