There are references to natural 20s and 1s in the first edition DMG. (The copy I'm looking at is the 1981 seventh printing, the last one with the efreet on the cover.)
The rules for saving throws specify "certain failure" for natural 1s. See page 81:
Certain Failure: As shown on the table, a 1 is ALWAYS a failure, regardless of magical modifiers to the contrary. However, as DM you may adjust such failures according to prevailing circumstances, although any adjudication which negates failure on a roll of 1 is not recommended at all. Another rule you may wish to consider is allowing a save (where applicable) on a natural 20, regardless of penalties.
But there isn't any concept of extra damage or other benefits for a natural 20, or extra penalties for a natural 1 (such as dropping or breaking your weapon). There isn't even an idea of a natural 20 being an automatic hit, for instance there's verbiage on page 70 ("Important Note Regarding "To Hit Adjustments") advising that to hit adjustments are made to AC, not to the to-hit die roll:
By so doing it is still possible for opponents to roll natural 20s and thus score hits.
There are special cases where rolling certain natural numbers leads to extra benefits, for some magic items (Vorpal Sword, Sword of Sharpness: lopping bits off) and monsters (the Purple Worm and Remorhaz: swallowing whole).
For the Sword of Sharpness, for instance, on page 166:
Sword of Sharpness is a weapon that is treated as +3 or better for purposes of who or what can be hit by it, even though it gets only +1 bonus "to hit" and on damage inflicted. Its power is great, however, for on a very high "to hit" die roll as shown below it will sever an extremity - arm, leg, neck, tail, tentacle, whatever - determined by random dice roll:
It lists a "modified score to sever" of 19-21 for normal opponents, 20-21 for larger than man-size, and 21 for solid metal or stone targets, that "considers only the sword's bonus of +1". (Can't help thinking it would seem clearer to list the natural numbers needed instead.)
In the Monster Manual (the copy I'm looking at is the 1980 printing, the most common one), the purple worm (page 80) and remorhaz (page 82) have special case rules for swallowing opponents whole, the word "natural" isn't used but seems implied, for instance:
A purple worm attacks by biting (2-24 points of damage), and any hit 20% over the required score (that is 4 or more over the required number) or a 100% (die roll of 20) score in any case indicates the creature has engulfed its victim.
None of the Basic D&D rules (Holmes, Moldvay, or Mentzer) make mention of natural rolls except for the monster and magic weapon special cases.
Second edition has an optional rule for this as described by John's answer. There was also a Player's Option: Combat & Tactics book that was published, which provides not one but two other critical hit systems (page 32), the simpler of which follows:
Critical Hits
Although critical hits are mentioned in the core AD&D rules as an option, this system works differently than any provided there. A critical hit may occur when a character rolls exceptionally well during his attack. The attack roll must be a natural 18 or higher, and the target must hit the target by at least 5. A 1st-level fighter with a THAC0 of 20 can achieve a critical against an opponent with AC 5, since he can roll a 20 and hit him with 5 to spare, but he can't get a critical against an opponent with AC 4 (or better).
As a basic rule, critical hits inflict double damage. However, Chapter Six is devoted to the topic of critical hits and presents an integrated system that accounts for the strength of the blow, the location injured, and a dash of luck.
If PCs can get critical hits with great attack rolls, monsters should be able to as well. Otherwise, the balance of the game shifts in favor of the player characters.
As it says, Chapter 6 features the second, more involved system, which takes effect with the same conditions as the simple system above but additionally when the target fails a save vs. death. There are a bunch more rules and charts with hit locations and specific injuries to roll for.
Third edition is the first occurrence in D&D rules of a non-optional critical hit rule. From the Player's Handbook, page 123:
Critical Hits: When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target's AC, and you have scored a threat. The hit might be a critical hit (or "crit"). To find out if it's a critical hit, you immediately make a critical roll - another attack roll with all the same modifiers as the attack roll you just made. If the critical roll aslo results in a hit against the target's AC, your original hit is a critical hit. (The critical roll just needs to hit to give you a crit. It doesn't need to come up 20 again.) If the critical roll is a miss, then your hit is just a regular hit.
There was mention of Empire of the Petal Throne as an early influence so here is its critical hit rule. The page number is 32 for the DW printing, 34 for the booklet from the 1975 TSR box:
731. Double Damage and "Instant Death."
If a player throws a 20 on the 20-sided die to hit (Sec. 7.20), he does DOUBLE DAMAGE. This must be a "natural 20": i.e., not including any hit bonuses. Note also that the character's damage bonuses are added in only once. Thus, for example, if a 4th level being hits a first level opponent shaking a natural 20, he rolls four dice instead of two and adds any damage bonuses to the resultant score.
A player who throws a "natural 20" also has the opportunity to try for an "instant death" kill: if he can follow his "natural 20" with another throw of 19 or 20, the opponent is instantly dead, whatever its hit dice may be. This simulates the "lucky hit" on a vital organ.
(This doesn't sound any worse than the 2nd edition D&D optional rule, but damage dice in EPT were calculated by cross-indexing the attacker's level vs the target's level, so if the attacker's level is much higher than the defender's this can be brutal.)
Although TSR published EPT at one point, and there was probably some cross-fertilization going on (Dave Arneson was one of the players in Professor Barker's game, one of his PCs was Captain Harchar), it was never branded as D&D and the rules were significantly different; I am not sure how relevant it is here.
Wikipedia makes a claim that EPT introduced the concept of critical hits, but that line says "citation needed". Before TSR took it up EPT was self-published in 1974.
One other book that might be relevant is the Blackmoor supplement to Original D&D, which includes an option for a hit location system, according to Wikipedia:
The book also introduced a section on diseases and a hit location system, wherein each individual body part of a character or monster was assigned its own hit points. The odds of hitting any specific body part changed depending on the character's height and weapon reach. If any individual body part was brought down to zero hit points, the creature would be crippled or killed.
This would pre-date any of the other references here except for EPT.
