The term “Basic D&D” covers quite a number of products and means different things to different people. I use the most common definition that refers to all of the products “between” Original D&D and Advanced D&D. The matter is complicated in that many of those products were not actually between those groups but instead were produced at the same time (or even after) AD&D. I’ll trust that the purists will forgive me if the lines are blurred a bit in the pursuit of clarity.
Holmes Basic Set
The original Dungeons and Dragons Basic Set was a box set published by TSR in 1979. It included the basic rule book, produced by Dr. John Eric Holmes, and a set of polyhedral dice. Depending on the printing, the set also included Module B1: In Search of the Unknown, Module B2: Keep on the Borderlands, or a set of geomorphs, tables and maps.
The rulebook covered character levels 1st-3rd, and incorporated concepts from the first three original D&D supplements: Grayhawk, Blackmoor, and Eldritch Wizardry. The book suggests that players who wish to progress past 3rd level should move on to the Advanced Dungeons & Dragons system (which was to be largely incompatible and not yet in print).
Moldvay Basic Set
In 1981, a major revision of the Basic Set (edited by Tom Moldvay) was published. Unlike the Holmes booklet, Moldvay designed this set strictly from the original booklets, choosing not to include anything from the supplements.
This set also included module B2: Keep on the Borderland and a set of dice.
Cook Expert Set
TSR released an Expert Set, edited by David “Zeb” Cook, in 1981. It was intended to be an expansion of the Moldvay Basic Set, allowing characters to progress to 14th level.
Mentzer Basic Set
A third revision of the Basic Set was made in 1983, edited by Jacob Franklin “Frank” Mentzer III. This set included two booklets, one for the DM and another for players.
Mentzer Expert Set
Frank Mentzer revised and expanded the Mentzer Basic set to create the Mentzer Expert Set. This set was released in 1983 and allowed characters to progress to 14th level.
Between 1983 and 1985, there were a series of five Mentzer sets produced (The aforementioned Basic and Expert rules as well as the Companion, Master, and Immortals rules) that allowed characters to attain progressively higher levels. These five sets are collectively referred to as BECMI, an acronym taking the first letter of each of the sets.
The Dungeons & Dragons Rules Cyclopedia was published by TSR in 1991, as a continuation of the Basic edition. It included rules from the various Basic Sets, Expert Sets, Companion Set and Master Set. This book, at least in theory, is a culmination of all the rules ever devised for original and basic Dungeons & Dragons.