Numerous core aspects of 5th Edition exist solely to encourage thinking in-character, including mechanical rewards for good roleplay and detailed character backgrounds with lasting effects in the game.
Bounded Accuracy is a core design ideal of 5th Edition. In short, it means the numbers don't get too big; this lets weak challenges remain relevant for longer, and strong ones become survivable sooner. In practice, it also places more focus on a character's inherent abilities and less on magical equipment than has been the case in recent editions.
Wizards of the Coast provides free PDFs containing the complete core mechanical rules of 5th Edition D&D (combat, exploration, spellcasting, etc.) and a selection of iconic character options, monsters, magic items, etc. drawn from the full game. Though short on variety, the basic rules are playable and accurate to the full game.
This is the most recent edition of the game and is the only one for which official content is being produced. Do not expect to see new official adventures or splatbooks published for any of the older editions of D&D.
Basic D&D is 6 ability scores, a few skills, a character's saving throws, some equipment and, if a spellcaster, some spells; and that's it. This makes the game more about what's happening in the game rather than an exercise in firing off various powers on your character sheet.
As with AD&D 1st Edition, Basic/Expert D&D is one of the favored editions of the Old School Renaissance community, who regularly offer wildly creative material for it and have published several retro-clone editions, all of which are currently available online either for free or for low prices. OSR blogs on Blogspot and Google+ communities serve as sources of considerable materials that can be used without modification for this edition, and some of these materials break far away from the "standard vanilla fantasy" flavor of Wizards of the Coast's publications.
Lamentations of the Flame Princess is a horror/fantasy game (not for kids). Fully compatible with Basic D&D material, it features a few non-breaking adjustments such as a vastly simplified encumbrance system.
2nd Edition AD&D presents a number of optional rules DMs can implement in their campaigns if they wish. Later books in the line, the Players' Option books and the revised core rulebooks, offered alternate rules to customize games according to preference. Some of these rules were incorporated into 3rd Edition, yet in 2nd, none are necessary to play the game.
2E causes players to re-think their decisions before engaging in risky behavior. Death can happen at any stage, and there won't be a chain of saving throws to stave off the reaper: You reach 0 HP, you're dead! Much of the focus, then, reverts to problem solving and role playing.
Due to concerns over public perception, for 2nd Edition TSR chose to eliminate mature themes and explicitly "evil" player options; remove references to angels, demons and devils; and otherwise smooth over potentially offensive content in the core game. This deliberate sanitization is easily reversed in personal play, but results in a presentation that feels somewhat naive and juvenile.
The Old School Renaissance community has a significant online presence on Blogspot and Google+, and features some of the most creative minds in fantasy gaming, all offering their ideas to the gaming audience for free or for generally low prices on PDF storefronts. As they aren't bound by Wizards of the Coast's need to cater to a mass market, they aren't obligated to present only a single "flavor" of fantasy. People playing 1st Edition (or one of its many available retroclones) will have no lack of material and inspiration to draw from.
Later editions were, to varying degrees, designed for market appeal; but the original AD&D is little more than one man sharing notes on his personal vision. It's tailored to his thematic tastes, and very often as much care is put into explaining the reasoning behind a rule as into defining the rule itself.
Due to countless situational modifiers, players and DM alike have to put a great deal of thought into decisions about position, order of actions, etc., to the point that six seconds (one combat round) of "in-game time" frequently takes half an hour or more of play time.
With so many options available to the player, and the almost unlimited ability to combine them, certain "character builds" are patently superior to others. Players who spend a lot of personal time poring over the rules can often create characters so powerful that players who don't optimize feel useless in games with those who do.
When Wizards of the Coast discontinued D&D v3.5 development in favor of 4th Edition, Paizo took advantage of the Open Gaming License to publish an updated revision of the 3.x rules under the title Pathfinder RPG, which is still their flagship product. Notably, Pathfinder RPG has consistently outsold 4th Edition.
The D&D v3.x family is designed around building characters, monsters, magic items, etc., from small simple pieces. For each piece, any of countless others can be substituted, allowing near-infinite customization and variety.
As a grid-based combat game, 4th Edition offers well-balanced and tactically rich gameplay. In particular, it features numerous effects which force or prevent movement, often preventing simple "charge and attack" tactics.
Compared to prior editions, 4th Edition D&D has dramatically simplified rules. There are very few special cases to remember, and all character classes (Fighter, Wizard, etc.) share a common pattern in how they gain and use combat abilities.
4th Edition downplayed simulation, and instead embraced cinematic action. The players are assumed to be the main characters of a story which revolves around them, rather than merely one faction in a far larger world.
Making it fun was such a priority in 4th Edition that it frequently trumped making sense. This manifests in ways ranging from "minions" who die in hordes at the lightest touch, to broken weapons simply repairing themselves.
Though most other editions of D&D assume miniatures are used for combat, they have all provided rules for playing without them. 4th Edition absolutely requires a grid, to the point that distances and ability effects are defined in terms of "squares."
To encourage boldness, characters are far harder to kill than in any other edition. This also allows players to invest heavily in the story and personality of their characters without facing the potential buzzkill of loss.