The Evolution of Magic

Men & Magic
Vol. I of the Little Brown Books takes the idea of the Wizard, or Magic User class of character and expands upon it for the new White Box. The original five named "levels" of  Chainmail Wizard, the Seer, the Sorcerer, the Warlock, the Magician and finally Wizard become more defined as a formal system of levels tied to experience is imposed on the class "Wizard" or Magic User as it is now called. The old names will reappear on the new list along with Medium, Conjurer, Theurgist, Thaumaturgist, Enchanter, and Necromancer to give a total of eleven titled levels. Presumably all additional levels of magic user continue to be known as Wizard.
I have always like the idea of the titled or named levels used in White Box. It has a quaintness which seems appropriate when introducing a character, PC or NPC as So-and-So the Conjurer, etc. Everyone instantly has recognition of the sort of character So-and-So is and how powerful they likely are. It also hints secret societies and membership rites and all the associated assumptions that might go with titles.
In addition to a system for characters to progress in levels, the LBBs also expands the list of spells and organizes them into levels. Chainmail is first and foremost a battle game and the spell list for its Fantasy Supplement consists of spells usable on the battlefield.  Men & Magic introduces a number of additional spells geared specifically for dungeon (and perhaps wilderness) adventuring. Additional spells not listed can be researched and theoretically at least any spell effect imaginable is possible.
The Fantasy Supplement in Chainmail lists spell complexity levels and the optional casting roll score needed varies by "level" of Wizard and complexity of spell. Men & Magic lists six levels of magic user spells (later expanded to 9 levels in Greyhawk), with the higher level spells usable by higher level magic users. Spells are cast without a dice roll for success, but are one-shot affairs which must be memorized and held in memory until cast. The 1st level Seer famously has a single 1st level spell they can memorize and cast each day.
Men & Magic therefore introduces the so-called "Vancian" magic system to fantasy gaming. The term "Vancian" is a reference to certain Jack Vance stories set in his Dying Earth cycle in which magic users commit a number of spells to memory prior to heading out on an adventure. Having memorized the particular spell, in the Jack Vance stories, the magic user can then cast it quickly, but in the process the spell memory is forgotten. Or as Men & Magic states:

Spells & Levels: The number above each column is the spell level (complexity, a
somewhat subjective determination on the part of your authors). The number in each
column opposite each applicable character indicates the number of spells of each
level that can be used (remembered during any single adventure) by that character.
Spells are listed and explained later. A spell used once may not be reused in the
same day.

White Box Magic Users may additionally create (or acquire) magic scrolls and potions containing any spell that they know (and which is recorded in a personal spell book). Scrolls and potions allow magic users to effectively cast additional spells above and beyond those available to them through memory. Spell books, potion ingredients, scroll materials and spell research all requires expenditure of in-game wealth by the Magic User and therefore helps support an in-game economic system and a reason for the Magic User to adventure.
Men & Magic lists Saving Throws by class and level for resistance to spells and other potential hazards. Spells which adversely affect an individual usually allow for that individual to attempt a Saving Throw (of the die) in order to avoid the worst. Some spells have variable effects such as fireball causing a variable amount of damage as thrown on the dice. Many spells in White Box magic require no dice involvement and just take effect when cast.
From these beginnings fantasy game magic continues to expand and evolve. Being a staple of the genre, magic appears in almost every FRPG. Being "magical", its effects vary widely and games have portrayed it in many ways. Game mechanics found outside White Box have been developed to govern at-the-table effects of magic using "magic points" which are spent by casting, there-by limiting spell use. The addition of rules for spell components and detailed casting time have added complexity to some Editions of the Game. World creators often "house-rule" certain aspects of magic for their setting. Keeping the "magic" feel in game magic is a discussion often had among many veteran gamers. What role magic plays in shaping a particular milieu is often one of the defining decisions a referee makes in defining his/her game. As is often the case, the more experience I have with the hobby, the more I appreciate the way things work in that original White Box game.