OD&D - The original game had only three classes (Cleric, Fighter, Magic User). Cleric spells up to 5th level, Magic user spells up to 6th level. Every attack except for certain monster abilities did 1d6 damage if it hit. There wasn't a lot of difference between characters in terms of combat capabilities. Characteristics didn't have many modifiers.
OD&D plus Greyhawk Supplement - The Greyhawk supplement transformed OD&D into a form of older edition D&D that is recognizable by most gamers today. Characteristics have more modifiers and exceptional strength was introduced. Variable damage dice for different weapons and creatures was introduced. The number of spell levels increased.
Holmes Edition, B/X D&D, Mentzer D&D - Similar to OD&D plus Greyhawk including selected elements from other supplements, with the rules rewritten for clarity and organization. Playing a Race meant playing a class. For example a Dwarf used only the Dwarf Class. Both B/X and Mentzer were divided in distinct books that focused on a specific range of levels. Later the Mentzer version was combined into the Rules Compendium. The biggest difference between these rules and AD&D was found in higher level play. Mentzer D&D had specific rules for running domain, mass combat, and even becoming a immortal i.e. god.
AD&D 1st Edition - OD&D plus Supplements plus Strategic Review articles are combined, rewritten, and organized into a three book set. One of the reason behind this edition was to standardize how D&D was played to make running tournaments easier. The most popular version of older edition D&D. Bonuses for characteristics roughly go up to +4 and are capped at 18 except for exceptional strength.
A lot of extra details are added in Gygax's distinctive writing style. Some section are poorly designed or understood like the unarmed combat rules, initiative, psionics, human dual classing, etc. While other are widely adopted, classes, races, spells, magic items, etc. Characters select a race and a class. Non-human race can multi class which involves splitting experience between multiple classes. Non-humans were generally limited to a max level (often low).
AD&D 1st Edition plus Unearthed Arcana - This version shifted the power level of the game upwards by allowing increased level limits for non-human, new classes that were slightly more powerful, and weapon specialization for fighters. Later AD&D hardback books (the two Survival books) expanded the use of non-weapon proficiencies as a skill system.
AD&D 2nd Edition - Still basically AD&D 1st Edition but the rules have been reorganized and rewritten for clarity. Some content like half-orc, demons, and assassins were removed or changed due to media pressure. Character customization was expanded by using non-weapon proficiencies as a skill system and by allowing characters to take kits that confer various benefits. Combat has been redesigned to overcome the issues with initiative and unarmed combat that were part of the previous edition of AD&D.
Because of the success of Dragonlance, much of AD&D 2nd Edition run was focused on customizing the rules for specific settings or themes. TSR released a lot of different settings like Dark Sun, Birthright, and others.
AD&D 2nd Edition Skills and Powers - Player's Option: Skill and Powers introduced several rule systems that allowed extensive customization of a character.
D&D 3rd Edition - The first edition created by Wizards of the Coast, 3rd Edition took the idea of Skill and Powers and developed a cleaner system for customizing characters by designing the classes so a level of one class can stack on top of another class. A single level chart was introduced and a each level a character could take a new class or add another level of a class they already had.
In addition feats were added to allow character to further customize their abilities. A true skill system was introduced and integrated into the game. The underlying d20 system worked by rolling equal too or higher than a target number and adding various bonus. This was used across the game in a standard way. Problems developed at higher levels as the number of options increased to the point where players had a tough time resolving their actions.
In addition, when various supplements were combined, characters could be built that were considerably more powerful than other combinations. This version was also noted for releasing the d20 system under the Open Game License, which ignited a vigorous third party market.
D&D 3.5 Edition - This edition featured only small changes to the core game (and was mostly-but-not-entirely compatible with books written for 3rd Edition), but had its own extensive line of supplements which magnified the role of feats, prestige classes, and multiclassing in character customization.
This version of D&D is still the baseline for many D20 games still in print and active development, notably the Pathfinder game by Paizo.
D&D 4th Edition - This edition is a completely new game with only a few game mechanics carried over from the 3rd Edition. It has a simple set of core rules and defines all character and monster abilities as exceptions which are described in standard terms. Higher level combat has been simplified, and class has been designed to have specific roles in combat. Every classes has a diverse set of combat options to use. The use of a battlegrid and miniatures is part of the core rules. Classes and monster generally have a high fantasy flavor. There are multiple ways to heal centered on a new mechanic called healing surges. Combat takes noticeably longer than any prior edition except perhaps for high level 3rd edition combat. While not present at the game's launch, this edition is noted for popular use of on-line computer tools, particularly an online character builder that integrates content from all the supplements. Wizards of the Coast originally intended to create a "virtual tabletop" as well, but the project was never completed.
D&D Essentials - This was an alternative set of core books for 4th Edition, with simplified classes intended for first-time players. Essentials was designed to be cross-compatible with 4th Edition, with different versions of the classes usable side-by-side.
D&D Next - This is the next edition of D&D currently under development. From what little we know the core will be similar to older edition D&D rather than the new system developed in 4th edition. It has been said by the designers that options will be present to allow the game to be played in a manner similar to 4th edition. The designers' focus is to make it modular so that a referee can construct a campaign that feels like their favorite edition.