Elisha Gray created single-note oscillators in the 1870s using electromagnetic circuits. Thaddeus Cahill created an electrical instrument in the 1890s. Lee De Forest invented the Audion piano around 1915 (using vacuum tubes). Instruments based on synthesis were developed in the 1930s and 1940s. Here's a recording of a Novachord from the late 1930s.

As for computer generated sound, Ada Lovelace (who died in 1852) suggested that the machines Babbage was developing could be used to produce music. The first computer to play music was CSIRAC, in the 1950s (check how that thing worked if you are into electronics and computers, from our modern context it's completely bonkers how things worked back then).

I can't help much with the mainstream music part (I mostly know about classical music, and computers). For classical music, people were experimenting with the instruments as soon as they were invented. The Novachord would be my guess for the first influence in mainstream music, but I can't really back that up.

I feel that /u/erus and /u/drylaw have given excellent explanations of the history of synthesisers and computer generated sound and when it first started to be introduced into mainstream music; I felt like the interesting thing I could add was about the point when synthesisers became ubiquitous, unremarkable parts of mainstream pop music the way that they are today.

Essentially, synthesisers in pop music until the late 1970s were either a) something of a novelty element (think Hot Butter's 1972 'Popcorn'), b) an interesting experimental sound used by progressive-minded bands to say something about the modern world (think Kraftwerk's 1975 'Autobahn') or perhaps c) an interesting texture to fill out a sound (e.g., the 1960s examples of Simon & Garfunkel and The Beatles). In contrast, synthesisers and drum machines in pop music by the mid-1980s were so ubiquitous that session drummers were going out of business-slash-frantically trying to learn how to program drums on computers.

There were a few factors that led to this change between the late 1970s and the early 1980s (as Bob Stanley's thoughtful history of pop Yeah Yeah Yeah: The Story Of Modern Pop argues). One was the availability of relatively cheap, relatively portable, relatively easy to use polyphonic synthesisers and drum machines - the likes of Casio's Casiotone series and Yamaha's Portasound series of synthesisers meant that you didn't need to understand what an LFO or a resonance filter was in order to operate a synthesiser, and that musicians could relatively easily change sounds between songs without too much fuss. This democratised the synthesiser. In contrast, the innovative use of synth sounds on early 1970s Stevie Wonder records required an enormous room-sized synthesiser, 'TONTO', operated by a couple of computer music experts, and a very patient layering of monophonic sounds.

Another change was that synthesisers and drum machines were a logical next step for people who believed in punk ideology. The punk movement of the late 1970s took pride in their lack of musical ability, believing that simplicity and energy and attitude resulted in better music than technical music ability and ambitious progressive epics. Once synthesisers became ubiquitous, the next logical step for many punks was to abandon guitars and drums for synthesisers and drum machines - who needs to learn how to play drums when you can much more easily call up a beat on a drum machine? As a result, early adopters of synthesisers in British pop inevitably seem to be former members of punk rock bands, the most obvious being Gary Numan.

Simultaneous to the movement from punk to new wave embodied in a Gary Numan, disco found itself rapidly moving towards electronic sounds around the same time. Take reliable soul/disco journeymen Earth Wind & Fire; their 1978 hit 'September' has a fairly standard densely layered 1970s disco sound, with no obvious synthesisers. In contrast, their 1981 hit 'Let's Groove' has a much more electronic sound, with synthesised bass, a prominent vocodered vocal and a video featuring someone playing a keytar. Donna Summer's single 'I Feel Love', produced by Italian producer Giorgio Moroder, is usually seen as ground zero for this change. Disco DJs found that audiences flocked to dance to the likes of 'I Feel Love', as the synthesised sound meant deeper bass and robotic rhythms made for more viscerally effective dance music. Additionally, the rise of synthesisers, drum machines, and computer-generated sound at this point, also democratised dance music - it was no longer necessary to have a large, highly competent band like Earth Wind and Fire in order to generate big-sounding dance music.

Great overview by u/erus - I'll add to the part about how synthesizers later spread into mainstream music, culled from readings from personal interest.

To follow up from early synthesis, in the 1940s & 50s we also see the institutions of various electronic music studios: a well known one being the Groupe de Recherche de Musique Concrète (GRMC) in Paris, helmed by Pierre Schaeffer and Pierre Henry, whose work on musique concrète proved influential on later electronic and avant-garde musicians. In Germany the work of Karlheinz Stockhausen and others in Düsseldorf and Köln on electroacoustic music was similarly important. Looking at later developments in German electronic music (later termed as "Krautrock"), both members of Kraftwerk and Can studied under Stockhausen - and both groups would find mainstream success in Germany in the 70s, arguably Kraftwerk more so, although Can's song Spoon was succesful due to its use in a TV series, and featured an early drum machine.

I'll backtrack a little here to look at the spread of synths in US pop music, probably closer to your question. A German pioneer in synthesizer building, Harald Bode, developed an early modular synthesizer and sound processor in the late 50s. In the US Donald Buchla and Robert Moog built on Bode's designs for their own models, with the Moog becoming best known for its early use in music (the Buchla apparently being even more difficult to control, esp. at the time). Moog's first prototype was built in 1964, with portable devices becoming available only later.

With the first Moog prototypes, the musicians Paul Beaver and Bernie Krause (duo "Beaver & Krause") played an important role in explaining how to play the synth to other musicians, as there were many difficulties with controling them enough for use in songs. Here's a quote from a recent interview with Krause (from Wire magazine nr 366) talking about selling the earliest Moogs at the Monterey Pop Festival in the 60s:

Paul [Beaver] and I - with our last few dollars because we spent everything we had to buy a Moog - had set up a demontration booth at the Festival. We were so poor we had to borrow an amplifier to demo the synth. But the upshot was that a lot of groups bought a synthesizer. We sold maybe 10 or 12 - The Monkees bought one, the Doors bought one, and so on. The problem was that most groups were so stoned they couldn't figure out how to patch the chords. [...] At the time, nobody could play the instrument but Paul and me - The Monkees never took the synthesizer out of its box. They didn't know anything about it: we just set it up a few sounds at their sessions and let them fuss around with the keyboard until they had what they wanted.

He also mentions introducing the Moog to the UK, selling models to Beatles producer George Martin (who was no beginner with electronic sounds) in '68, to George Harrison in '70, and working with one with Mick Jagger in the early 70s. Apart from the technological difficulties, we see here the interest of popular artist wanting to incorporate new sounds into their music, probably to stay ahead of the competition. This becomes appartent in the already mentioned Monkees' recording the first album with a Moog in '67, with songs using the instrument by the Rolling Stones and the Doors (see Strange Days) following the same year. In the '68 we can see the Moog synth going from a novelty to becoming really embraced by mainstream music, with features by the Beatles and Simon & Garfunkel and its reach spreaching further through the portable Minimoog. As an aside, synths started being used more extensively in this time in experimental music at the time as well, with avant-garde Jazz pianist Paul Bley and free-jazz visionary Sun Ra using synths very early on to great effect (here's a nice short article on how Sun Ra received a Minimoog prototype from Robert Moog).

To close up, this introduction into mainstream led to synths becoming more ubiquitous in the 1970s', especially with new forms like Disco and Hip Hop using electronic sounds more heavily. The early 80s are then usually described as a breacktrough time for electronic music, with synths becoming available to a larger public and with the development of MIDI. To come back to the German influences I mentioned before: Various German Krautrock/Kosmische groups would become important for US electronic and Hip Hop artists, with esp. Kraftwerk's music being heavily sampled by Hip Hop pioneers like Afrika Bambaata.