Without further ado, here's a brief rundown of the progression of Classic D&D...
Original Dungeons & Dragons (a.k.a. OD&D, the "little brown books," LBBs)
The original bare-bones version of the game as set down by Gygax and Arneson. It features only three character classes (fighter, magic-user, cleric.) Race and class are considered separate, although hobbits/halflings and dwarves can only be fighters, and elves can function as either fighters or magic-users. Ability scores are much less important than in later editions. Strength, Intelligence, and Wisdom serve only as prime requisites for the three classes and determine characters' XP adjustments. Very high or low Dexterity modifies missile fire by + or -1. Very high or low Constitution modifies hit points by + or -1 per die. Charisma determines number of hirelings and their loyalty. All weapons do 1d6 damage, and all classes used d6 for rolling hit points.
Supplements introduced the thief class and other rules modifications.
A rules set edited by J. Eric Holmes, which compiled the original game plus supplemental material into a more accessible format. Character classes use different dice to determine hit points: d8 for fighters, d6 for clerics, and d4 for magic-users and thieves. The lines of race and class begin to blur for demihumans; while it's stated that dwarves and halflings function as fighting men and elves as fighting men and magic-users, the elf and halfling are assigned a d6 hit die.
Initiative in Holmes is in descending order of Dexterity scores, with ties broken by a d6 roll.
It also features a unique five-option alignment system: Lawful good, lawful evil, chaotic good, chaotic evil, and neutral.
Holmes Basic apparently was originally intended as an introductory set which would prepare players for the upcoming Advanced Dungeons & Dragons game. It provides for character advancement only to level 3; for higher levels it directs readers to the AD&D game.
B/X, short for Basic/Expert, a.k.a. Moldvay/Cook
It was with this rule set, edited by Tom Moldvay (Basic Set) and David Cook with Steve Marsh (Expert Set) that classic D&D really became a game unto itself, separate from both OD&D and AD&D. The Basic Set lays out the process of creating characters and details their advancement from level 1 to 3. It also describes dungeon adventures. The Expert Set details character levels 4-14 and provides rules and procedures for wilderness adventures. A Companion Supplement is mentioned which would allow advancement to level 36, but this never materialized.
The B/X rulebooks feature cover art by Erol Otus, and interior illustrations by Otus, David S. LaForce, Jeff Dee, Jim Roslof, and Bill Willingham. The style of the artwork is a very strong influence in the overall feel of the books.
Among the innovations of B/X are:
Race-as-class demihumans. Dwarves, elves, and halflings are explicitly presented as character classes unto themselves.
Expanded game effects for ability scores. Every ability score now affects every class, rather than only providing an XP adjustment to the classes who have it as a prime requisite. Strength modifies melee attack and damage rolls, Intelligence determines number of languages spoken, Wisdom modifies saving throws vs. magical effects, Dexterity applies to missile attack rolls and Armor Class, Constitution modifies hit points, and Charisma affects NPC and monster reactions as well as retainer numbers and morale.
Standardized bonuses and penalties for ability scores. While the original and Holmes editions featured a different scale of bonuses and penalties for each ability score, and AD&D took this to even greater lengths with different scales for each element that the ability score modified, B/X implemented a standard scale:
d6 initiative: Each side rolls 1d6 at the beginning of each combat round; the side with the higher roll acts first. Ties are considered simultaneous.
Morale: Each monster is assigned a Morale score, which represents its courage in battle. Morale is checked at certain points in a fight by rolling 2d6 and comparing it to the creature's score. A roll higher than the Morale score indicates that the creature will try to flee or surrender, while a lower roll means it fights on.
Variable weapon damage: Moldvay Basic provides an option for weapons to do differing ranges of damage, ranging from 1d4 for daggers and clubs to 1d10 for two-handed swords and polearms.
BECMI, an acronym for Basic, Expert, Companion, Master, Immortal; a.k.a. Mentzer D&D.
Rather than proceeding with the B/X Companion Supplement, TSR rebooted the line with a series of rulebooks edited by Frank Mentzer. Substantively, the Mentzer Basic and Expert rules are virtually identical to Moldvay/Cook, with a few minor changes and clarifications and a few tweaks and omissions from the monster list. Advancement of thief skills, which in B/X maxed out at level 14, is modified in order to stretch it over 36 levels.
The primary difference between the two editions is in presentation and tone. The Mentzer Basic rules include lengthy tutorials for new players, and are considered to be more "kid friendly." Cover art by Larry Elmore and interior illustrations by Elmore and Jeff Easley contrast sharply with the weird and stylized art of the B/X illustrators.
The Companion Set, being an entirely new book rather than a fresh iteration of an existing rule set, departs much more radically from the substance and style of B/X, featuring complex systems for adjudicating unarmed combat, mass combat, and dominion management. The systems add entirely new stats and mechanics which feel foreign to the simplicity of B/X D&D to many B/X enthusiasts. Unlike its never realized predecessor, the Mentzer Companion takes characters only to level 25, reserving the remaining levels for Master. The demihuman level limits prescribed by the Basic and Expert rules (both B/X and Mentzer) are circumvented somewhat through the use of "Attack Ranks," which improve the character's attack rolls while providing none of the other benefits of levels such as additional hit points and improved saving throws.
The Master Set extended character advancement to the established maximum of level 36. It also includes a complicated system of weapon mastery, wherein characters may choose to increase their skill with specific weapons, resulting in vastly increased damage and various other offensive, defensive, and miscellaneous benefits. The War Machine mass combat system from the Companion Set is supplemented with Siege Machine rules, another layer of complexity. Methods for characters to pursue and win immortality itself are discussed as well, culminating in the Immortal Set, which essentially allows the campaign to proceed with PCs as gods or demigods.
BECMI spawned a plethora of supplements and sourcebooks, including the popular Gazetteer line which explored the lands and cultures of the Known World setting first presented in the Cook Expert rulebook. Besides setting detail, the Gazetteers and supplements such as the Creature Crucible series added a vast array of new classes and subclasses, spells, monsters, and a system of General Skills, similar to AD&D 2e's Nonweapon Proficiences but more freewheeling.
Most adventure modules published for this edition did not make use of optional rules such as Weapon Mastery and General Skills, and so were pretty much fully compatible with the old Moldvay/Cook rules. In fact, most published materials for Moldvay/Cook, Mentzer, or Rules Cyclopedia D&D are for all intents and purposes completely compatible, which probably accounts for the perception that Classic D&D is all of a single edition.
Essentially the Mentzer rule sets, compiled into a single hardbound volume by Aaron Allston. The cyclopedia makes the General Skills rules from the Gazetteers rulebook official, although still described as optional. The artwork has neither the grandeur and charm of Elmore's work in the Mentzer edition, nor the weird/creepy vibe of Otus and Co. in the Moldvay/Cook rulebooks.
Many clone rule sets have been published. Though none exactly duplicates the editions above, each generally hews closer to one set or another, and is often explicitly designed to emulate the feel and style, if not the exact rules, of a particular edition.
Swords and Wizardry emulates OD&D.
Labyrinth Lord approximates B/X, with a few notable differences in equipment lists, cleric turning tables, and the fact that clerics receive a spell at 1st level, unlike B/X.
Dark Dungeons resembles Mentzer edition D&D.
Basic Fantasy RPG has elements of B/X, but separates race and class.