Distortion/Overdrive
The terms "distortion" and "overdrive" historically came from describing the output of a loudspeaker driven by tube amplifiers. Strictly speaking, "distortion" can refer to anything that changes the input in any way other than amplitude adjustment. As a guitar effect, "distortion" in the modern parlance is reserved for gross distortion that changes the timbre of the instrument radically, adding overtones that are not present in the input signal. "Overdrive" still means what it originally means, namely clipping some amplifier stage somewhere along the signal path, and the term is generally reserved for milder changes in timbre than "distortion." "Overdrive" adds enough overtones to excite the ear, but not enough to lose the original input timbre. It is a very subjective distinction between the terms "fuzz," "overdrive," and "distortion."
Op Amp Distortion and Overdrive Topologies
History
In guitar effects units, the terms are not commonly used until about the mid 1970s. Early effects units were called "fuzz boxes" or "boosters." Commercial guitar distortion effects begin with 1962's Maestro Fuzz Tone, which does grossly distort the signal, but the Fuzz Tone lacks sustain. Over the next decade, the fuzz box would evolve from a percussive novelty effect into an essential element of the electric guitar sound.
First appearance of "Distortion" on an effect unit appears to be the Vox Distortion Booster in 1965. "Drive" appears on the Univox Uni-Drive in 1971. Also in '71 is the Colorsound Overdriver. "Overdrive" and "Distortion" have a break out year in 1974 with the Lyle Distortion Sustainer, Maxon OD-801 Distortion Sustainer (D&S), and Ibanez OD-850 Overdrive. Notably, all three of the 1974 effects are based on the Electro Harmonix Big Muff Pi. The Nu-Fuzz Distortion Blender is another early use of the term, but I haven't determined the year this effect first appeared.
Upgrading from Single Transistors to Op Amp ICs
On the technical level, the units that became known as distortions and overdrives embraced diode clipping over all other methods. The earliest examples used simple discrete transistor amplifiers, usually just single transistor stages, and then clipped the output passively. The problem with the single transistor stage is that overall gain is very limited with just one transistor. Trying to get a single transistor to provide a lot of gain tends to sound more like fuzz than drive.
By the early 1970s, integrated circuit technology had advanced to where operation amplifier (op amp) "chips" were cheap enough to use in guitar effect boxes. The early op amp designs were mostly limited to the uA741, the MC1458, and their copies. In 1974, the RC4558 was launched, and this became one the most popular audio op amps. Op amps contain dozens of transistors all internally connected to form a high gain amplifier, all within a single package roughly the size of a 1960s single transistor. When the diode clipping effects replaced single transistor stages with high gain op amps, the modern overdrive/distortion was born.
Defining Distortion
Early contenders for the first op amp distortion/overdrive are the Dan Armstrong Blue Clipper (UK/pre-Mutron) and the MXR Distortion +. Armstrong claimed in interviews that MXR copied his product (unconfirmed). The MXR Distortion + is also a contender for defining the modern idea of what a "distortion" pedal sounds like.
Defining Overdrive
The Ibanez Tube Screamer TS-808 from 1979 is a contender for defining the "overdrive" sound. The Tube Screamer has a wide range of subtle drive and boost, only reaching "distortion" territory at its maximum settings. Bill Finnegan's love of the Tube Screamer led him to create the Klon Centaur in 1994, which has become cult phenomenon for its ultra subtle overdrive sound.
Boss' Unusual Move: From Op Amp ICs to Discrete Op Amps
Boss released an IC based "Overdrive" model called the OD-1 in 1977. They made the rare move of reverting back to discrete transistors, in a primitive op amp arrangement, when they released the OD-2 "Turbo Overdrive" in 1985, and kept releasing discrete designs with the OD-2R (1994) and OD-3 (1997).
Op amps were not the only ICs to find there way into distortion effects in the 1970s. CMOS logic chips were also pressed into service. CMOS stands for "complimentary metal oxide semiconductor" and these chips are made up of MOSFETs. The (dubious) belief that MOSFET devices would somehow be sonically similar to vacuum tubes led to the design of the CMOS chip distortion topology that is still popular today. The CD4049 and CD4069, both "Hex Inverters" (6x inverting amplifiers per 14 pin chip) are the usual suspects. The Electro Harmonix Hot Tubes (1978?) is a commerical example. Craig Anderton's Tube Sound Fuzz, a DIY project, may have been the first widespread version after it was included in his 1975 book "Electronic Projects for Musicians." I have seen many DIY and boutique pedals that were Tube Sound Fuzz variations inside.
Tube Hybrids
As solid state technology in power amplification grew and challenged tube amplifiers, musicians and audiophiles debated the inherent timbres of the two technologies. This carried over into the overdrive/distortion world. Many guitarists rejected solid state effect units as harsh sounding compared the overdrive achieved by plugging straight into a tube amplifier and turning the controls up. This created a niche for tube based overdrive/distortion guitar effects. Usually these things are not solid state free, but the presence of the word "tube" in the graphics and a glass vacuum tube sticking out of the enclosure is usually enough to convince an end user that they are hearing "tube sound."
The BK Butler Tube Driver may be the earliest such unit, developed around 1978. The Tube Driver does contain an op amp distortion circuit, which is then passed through a tube. The overall sound is very op amp distortion to my ears.
Tube Distortion Pedals
Electro Harmonix released an interesting line of tube pedals in the late '90s/early '00s. These are quite notably for mostly eschewing all solid state components besides diode rectifiers for the power supply (at least in the few that I've peeked into...).
The designer's website is quite a trip: http://www.puretube.com/
The large and awkward sizing along with odd power supply requirements and very odd power supply connectors may have limited this line's potential among the general public. Although EH still sells these pedals, they may just be old stock. No new designs have appeared for a long time.
Pedals that offer overdrive and distortion as two separate modes
The modern distinction between overdrive and distortion is often simply whether the clipping diodes are within a negative feedback loop (OD) or just passively shunt clipping (Dist).
This list will surely grow as I find more examples, but here are some pedals that go as far as to offer selectable OD/Dist:
Maxon OD-01 The Driver
Fulltone Plimsoul (Stage 1/Stage 2)
Return to discrete transistors
I've noticed several boutique pedal makers have embraced discrete transistor designs.
ZVex's distortion pedals seem to all be based around BS170 discrete MOSFETs with very simple 1 transistor stages cascaded together.
Devi Ever has many discrete transistor designs.