It all started with the Roland TB-303, and it was kind of a mistake. Roland initially designed this bass synthesizer to replicate a bass guitar and well, it wasn't that great at it. After just two years in production (1982-84) Roland stopped making the 303 and it seemed destined for obscurity until some guys in Chicago started finding them on the market at rock bottom prices.
For young producers gear like this was expensive so many of them jumped at the chance to get a Roland synth in their set-up. This synth was considered to be junk by many electronic musicians at the time but many producers from Chicago, including DJ Pierre and his Phuture group, found the secret potential of its signature squelchy sound that worked really well with House music.
The sound of the 303 became known as Acid, and Phuture's (DJ Pierre, Spanky and Herb J.) 1987 release "Acid Tracks" was one of the first tracks to really hit with this signature sound. Acid House was born and quickly spread through the clubs in England and Europe.
It wasn't long before literally every genre of EDM at the time was using the sound of the TB-303. From Acid Techno to Acid Trance to Acid Breaks the sound was and still is firmly embedded in the world of electronic dance music.
The prices of the TB-303 skyrocketed and it quickly became one of the most sought after machines on the market.
There were even specific mods like the Devilfish that came around in the early 90s used by guys like Richie Hawtin to pitch up the squelch to give it an even more definitive sound. You can hear the Devilfish being used on his album Artifacts (bc), listen closely for it at the end and you will hear the pitched up modification to the acid sound (example below).
Here are some classic Acid House tracks from the late 80s by Phuture (produced by Marshall Jefferson), Farley Jackmaster Funk, and Jack Rabbit, that kick-started the Acid craze.