All of the retro-clones and neo-clones mentioned below are games with which I have some familiarity. Many of them I've played. The others I've read. All of them have something interesting to recommend them.
There was a time when Wizards of the Coast hadn't made old versions of D&D available easily, and the retro-clones were very important then. Even now some retro-clones or neo-clones have better presentation than the original rules (Old-School Essentials Classic Fantasy) or present a simpler version of classic rules (Labyrinth Lord: Revised Edition with Advanced Edition Companion for AD&D), or provide additional details and further development of a particular D&D version (Adventurer, Conqueror, King System).
And, of course, if you are playing with kids who might not have the money to buy PDFs of old versions of D&D, many of the retro-clones are available in PDF or online legally for free.
What do you want?
Want races separate from classes and the most of the extra classes, races, and many spells from AD&D 1E, all on the 1 to 14 level scale of the classic Basic/Expert D&D (B/X)? See Old-School Essentials Advanced Fantasy.
Want to play classic AD&D 1E modules with all races, classes, and most spells from AD&D 1E, on a mostly 1 to 20 level scale, with 9th level magic user spells and 7th cleric spells, but with simpler mechanics? See Labyrinth Lord: Revised Edition & Advanced Edition Companion, or Advanced Labyrinth Lord.
Want something almost like AD&D 1E? See OSRIC.
Want something that feels like Original D&D, just what was in the three original Little Brown Books? See Swords & Wizardry Whitebox, Labyrinth Lord's Original Edition Characters, Delving Deeper, or Fantastic Medieval Campaigns, especially if you want a Chainmail-style system for individual and mass combat. Do you also want the thief class? See Swords & Wizardry Core.
Want something that feels like Original D&D, just before AD&D 1E came out, with the races and classes (except for gnomes and illusionists) and spells familiar from AD&D 1E, but simpler, on a mostly 1 to 20+ level scale, with up to 9th level magic user spells and 7th level cleric spells? See Swords & Wizardry Complete.
Like the 1 to 14 level scale and the lower hit dice of Basic/Expert D&D, but want more detailed rules, especially magic research, economics, and domain play (the D&D end game, where the players build strongholds and rule), with rules for building campaign specific classes and races, and don't mind some slightly different mechanics? See the Adventurer, Conqueror, King System.
I think I looked at Basic Fantasy first, but it wasn't what I was looking for at the time.
All its PDFs are free, and the print on demand books are very inexpensive.
I notice that the character class tables all go to level 20. I wonder if this is a residual artifact of the D&D 3E character classes going to level 20, since it doesn't match the B/X model that that is implied by the Basic Fantasy website with the statement “The Basic Fantasy Role-Playing Game is a rules-light game system modeled on the classic RPG rules of the early 1980's.” Compare that to Labyrinth Lord's rescaling of B/X to match AD&D.
Original Edition Characters
Labyrinth Lord: Revised Edition & Advanced Edition Companion
While Labyrinth Lord: Revised Edition (LLRE; PDF and POD) has been criticized for not being a completely faithful retro-clone of the Moldvay/Cook/Marsh Basic/Expert D&D rules (B/X), I think it still holds a useful spot. The changes that were made in the revised version of Labyrinth Lord made it easier to be used with Advanced Edition Companion (AEC). It adapted the 1–14 level scale of B/X to the scale of AD&D 1E, generally about 20 levels, and with AEC it added the separate races and the additional classes, spells, monsters, and magic items of AD&D 1E.
Advanced Labyrinth Lord
Advanced Labyrinth Lord compiles the information from Labyrinth Lord: Revised Edition and Advanced Edition Companion into one convenient book (ALL, PDF and POD).
Swords & Wizardry
Swords & Wizardry is inspired by the original D&D booklets and associated magazine articles. It comes in various versions, depending on how much of the supplemental material you want.
I find the original D&D booklets very interesting from a historical standpoint, but find Swords & Wizardry easier to use at the table.
Swords & Wizardry White Box
This is inspired by the original three booklets of D&D, so it includes only the Fighter, Magic-User, and Cleric, so it strips away everything but the absolute essentials.
Swords & Wizardry Core
This adds the thief class.
Swords & Wizardry Complete
If you are looking to play something with the feel of AD&D 1E, but want something even simpler than Labyrinth Lord: Revised Edition and Advanced Edition Companion, this is an excellent choice, as long as you don't need half-orcs, gnomes or illusionists.
I like how it discusses different ways to interpret the original rules, often giving more than option for use with S&WC.
Old-School Essentials (OSE) is published by Necrotic Gnome in two flavors so far, Classic Fantasy and Advanced Fantasy.
In my opinion, OSE has improved the clarity, organization, and presentation of its rules over the already very good Basic/Expert D&D rules. As well as clear wording it uses layout and typography to enhance its presentation of the rules, from the use of one and two page spreads that completely cover one subject to the careful use of bold and bullet points to call out important information, OSE really shines. The adventures that Necrotic Gnome have produced for it (The Hole in the Oak, Winter's Daughter, The Incandescent Grottoes, The Isle of the Plangent Mage, Halls of the Blood King, and Holy Mountain Shaker) continue this excellent use of layout, bold, and bullet points to present their information clearly and without the dreaded “wall of text” or “read aloud text”.
Old-School Essentials Classic Fantasy
Old-School Essentials Classic Fantasy (OSECF) is the best organized and presented version of the classic rules from the Moldvay/Cook/Marsh Basic/Expert D&D sets. This is the game that I wish had been around when I started playing. There is an SRD (OSESRD). I particularly like the Classic Fantasy Rules Tome, which collects everything into one substantial book, but there is also a version split into multiple books covering separate sections of the rules, so that the magic-user can look up his spells while the fighter is looking at the combat rules.
Old-School Essentials Advanced Fantasy
Old-School Essentials Advanced Fantasy is the best organized and presented version of the ideas of the classic rules from 1st edition AD&D, scaled to match the 1–14 level scope of the Moldvay/Cook/Marsh Basic/Expert D&D sets. If I was starting a new D&D campaign these are the rules I'd use, all things being equal.
Adventurer, Conqueror, King System
The name of this system is usually abbreviated as ACKS. One of its explicit design goals for to supply more rules for the end of game of Original D&D, where the players built strongholds and ruled, hence the name. It also has a more detailed economic system. It is more complicated than OD&D or B/X, but in a completely different way than AD&D 1E.
It uses some different mechanics, especially for “to hit” roles, so it might be considered a Neo-Clone, but it is inspired by B/X D&D, and keeps to the same 1 to 14 level scale of B/X. I particularly like its rules for building your own campaign specific classes from the Player's Companion.
I looked at OSRIC (OSRICKnK), (a free OSRIC PDF is available, as well as a free PDF of the Usherwood Publishing OSRIC Pocket SRD version) but was more interested in Basic/Expert D&D retroclones at the time. I think it is a well written, organized, and presented restatement of AD&D 1E. The Black Blade Publishing print edition is a very nice offset printed volume with a sewn spine. The Usherwood Publishing OSRIC Pocket SRD (PDF, POD) is quite affordable. I just wish OSRIC wasn't missing some of the classes. (I miss the Monk in particular.) I was also surprised by how many of the little quirks and restrictions of AD&D 1E have been ironed out, although many remain. Usherwood Publishing has some supplements that add some of the missing classes, though I'm not sure I like all their versions.
This is a retro-clone of The D&D Rules Cyclopedia.
Dark Dungeons does have changes from the Rules Cyclopedia, integrating some optional rules, extrapolating, clarifying and adjusting other rules, and integrating rules for Immortals. It does not include the Mystara setting and cosmology elements. The monsters differ somewhat.
Delving Deeper (Delving Deeper Description, Delving Deeper PDFs) works very hard to be as faithful to the 3 Little Brown Books of Original D&D (along with the relevant partsof Chainmail, with which they were intended to be used) as possible. The Delving Deeper V4 Reference Rules Compendium is a Print-on-Demand version of the three booklets. There is also the Delving Deeper SRD. I was particularly interested in Version 5 of Delving Deeper, as that promised annotated versions of the three booklets, but only one of the annotated versions has been completed so far.
Fantastic Medieval Campaigns
Fantastic Medieval Campaigns (FMC) is a retro-clone of the three Little Brown Books of Original D&D and Chainmail, available for free in PDF and in print on demand. I gather it was something the author did just so they had something easier to follow when playing OD&D with friends. I'm glad they made it available.
Unlike a lot of clones of OD&D, FMC, as well as including the familiar d20-based combat system, also includes a clone of the Chainmail rules, which can be used either for individual combat or mass combat. FMC is very well organized, and could be used by someone new to D&D to play in a group where everybody else is using the OD&D rulebooks, or as the rules for a whole group.
Basically, FMC is a very well written and organized restatement of Chainmail and the three LBB. The layout and design are uncluttered; the art is charming, the organization is superb, and the writing is clear. The physical book is an excellent example of the possibilities of print-on-demand, with the different sections of the book printed on different colored backgrounds, making finding specific sections much easier. The table of the contents at the front of the book is reasonably detailed, and each section has its own table of contents as its first page, opposite the title page for the section. The book has a glossary and indexes for monsters, spells, and tables.
In the end this is an exceptional restatement and clarification of the original game.
There are also retro-clones of games other than D&D, if you go looking.