Okay, you're designing an RPG, and you've got your Fighter, Mage, Thief, but something's missing. The magic adds a lot of strategic elements to the combat system, and you don't want to have just one magic user. You want to add more variety. But you can't just create a copy of the mage; it has to be different somehow. So how to make the two mages distinct? Simple. Have one attack and the other heal!
Black and White Magic is a common way of having more than one magician character or class while having them remain distinct. One of the mages specializes in offensive magic which causes damage, negative status, or otherwise harms the enemy, while the other specializes in defensive magic to cure their allies, remove negative status, or increase their own stats. They are often referred to as "wizard" and "cleric" (or some variation) respectively.
Not to be confused with Black Magic and White Magic, which are distinct tropes and don't usually overlap with this. If a character can do both, then he's The Red Mage. Characters who use black magic are Black Mages, characters who use white magic are White Mages.
WarCraft: In the second game, Humans got White Magic in the form of Paladins. Their counterpart used Black Magic, but both also had offensive casters. In Warcraft III, all races except the undead have an offensive caster and a defensive/healer, though the Nightelven Druid of the Claw could also transform into a bear, the heaviest ground unit on their side until the expansion. Said expansion also added a third spellcaster into the mix which specialized in Anti-Magic.
In World of WarCraft, pretty much all of the healer classes also have the ability to do damage, but special mention goes to the priest class, which comes off as the stereotypical "white mage", but also has access to some of the nastiest shadow spells in the game.
Final Fantasy is the Trope Namer. Since the first game (and intermittently thereafter), it has divided its magic into "Black" and "White" categories, with black mages getting elemental attacks, white mages getting healing spells, and red mages got both at weaker levels. Later games have added other varieties of magic, such as Summon Magic and time magic, but every game with a class system (and many without) has had its Black Mage and White Mage.
Funny enough, the original Black Mage wasn't just about hurting people, he also had a great selection of Status Effect spells. Due to damage progression, his endgame role was to buff the physical attackers who could deal far more damage that way than the mage himself. Later games gave these spells to White Mages before budding all the status magic off into its own sub-category (Green Magic).
Only true in regards to stand-alone boss battles where spell resistance and lack of multi-effect spells meant that a Knight hitting 32 times or so was more effective than nuking the boss with Ultima. Due to lack of attack spread by physical classes, the Black Mage was still useful in later dungeons because the high level damage spells targeted the entire battlefield, allowing the player to clear every enemy at once, rather then whittling away at the enemies one by one.
The Dragon Quest series uses this as well, starting with its third installment. It's unusual in that "clerics" not only specialize in healing, but also wind magic.
Ultima III: Exodus divides its spells into Wizard and Cleric, and has several classes which can use one or both to various extents. It gets a bit strange, since Wizard magic includes a spell that lets the caster use Cleric magic. Note also that this is the only game in the Ultima series which does this; all the others have only a single spell list which all spellcasting classes use.
Fire Emblem has two varieties of spellcasting equipment: books, which act as weapons, and staves, which generate healing and support effects.
In this case, the divide is extreme; the Cleric can't attack at all until promoting. He or she has to promote into either The Red Mage (Sage) or War Cleric-War Monk (An Axe to Grind).
Secret of Mana has two spellcasting party members: the Girl, who uses healing and support effects, and the Sprite, who uses attack magic.
Lucca and Marle from Chrono Trigger have this dynamic even before learning magic. Lucca starts with the Flame Toss technique, which involves throwing a line of flames at an enemy, and Marle has the Aura technique which restores HP.
The Might and Magic series has both Dark and Light magic as classes of spells that spell casting characters can learn. Light magic is mostly for improving your stats, defenses, and healing, while Dark magic is full of very powerful damage spells. In most of the games, any caster could learn both types of magic, but in Might and Magic VII you had to choose either the light or dark path, which limited you to using only one of them.
The Dungeon Siege series has Nature Magic and Combat Magic. Nature Mages aren't completely limited to healing and buffing though - they can cast offensive Ice Magic spells (for freezing and slowing effects) and are pretty good at summoning creatures. However, in sheer destructive power they are no match for a skilled Combat Mage.
The Avernum series has magic split into mage spells and priest spells, which are exactly what they sound like.
While healing falls exclusively under priest spells, mages also do more than a bit of buffing and priests can do nuking with Divine Fire and other spells. At high levels, a priest can be more dangerous in combat than a mage, but this may be Gameplay and Story Segregation, because all the most crazy-powerful characters in the setting are magi.
Dungeons & Dragons is most likely the Trope Maker and definitely the inspiration for Final Fantasy, above. It has always had the Wizard class, which specializes in blasting spells, and the Cleric class, which specializes in healing spells. The third edition furthered this split, dividing all spellcasting classes into Arcane and Divine casters.
In addition to Arcane and Divine casters, Fourth Edition also has Primal casters (which cover nature-based classes like the Druid). Furthermore, it's now possible to have Divine "Black Magic" users (such as the Invoker) or Arcane "healers" (like the new version of the Bard).
In addition to all this, the Divine casters themselves can be split into white and black magic. Evil-Aligned characters can take inflict spells instead of heal, which hurt people and heal undead.