The ship's computer in Star Trek: The Next Generation produces complex beeps/bloops, particularly when a user is interacting with it.
How were these sound effects created?
Here's the most relevant quote from the article that NKCampbell linked to in the comments (emphasis added):
'I have a work station here with a Synclavier, Tascam DA-88s, a little mixer and a patchbay,' adds Wolvington. 'It's basically an identical system and database to the one that Tomi Tomita works on in California, and I therefore either FedEx Tascam DA-88 8-track tapes with the effects laid down, or I modem sequences that allow Tomi to recreate what I've done.'
This strongly suggests all the sound effects were created with a Synclavier. Here's the first line from the Wikipedia page (emphasis added):
The Synclavier was an early digital synthesizer, polyphonic digital sampling system, and music workstation manufactured by New England Digital Corporation of Norwich, Vermont, USA.
Musical instruments have been used for sound effects for about as long as there have been sound effects, perhaps the most famous example being the Theremin. Synthesizers in particular have used for various sound effects for film and TV since their invention, especially for science fiction. One surprising example of synthesizer based sound effects is the little-known fact that for commercials in the late 70s and 80s, the sound of a Coke can opening and fizzing was created on a synthesizer by Suzanne Ciani.
Anyway, a Synclavier is a synthesizer which means it can make all kinds of beeps and bonks and whooshes and even long ambient thrumming and droning and hissing noises. I'm not sure how R2-D2 sounds were created, but it all could have been done with an analog subtractive synth. The Synclavier is digital which means it could be a bit more tricky to use and program, but it also had capabilities far beyond a typical subtractive analog synth (like the most famous synth ever, the Moog Minimoog).
A Tascan DA-88 is an 8 track digital tape recorder, which means they could layer 8 different sounds at once onto a tape. That's important for things like doors and other complicated devices where you might need a swoosh and beep sound at the same time when your synthesizer can only make part of the sound at once.
They had two identical setups, which means they could ship tapes back and forth and add to them, or "modem sequences". Here, "modem" is being used as a verb to mean "send using a modem". Back in the 80s and 90s, a modem wasn't a thing that connected you to broadband. It was a computer peripheral that let you make a phone call to connect to another computer that also had a modem. The speeds were very, very slow by today's standards, so it wasn't practical to send actual audio files over the modem. Instead, what they would send were MIDI sequences. A MIDI sequence is a set of MIDI instructions that would program the other Synclavier to make the same sounds at the same time as the original Synclavier. So one person could set up sound effects and perform them at particular times and record both the setup and the performance in a MIDI sequence, and then transmit that sequence via modem to the other studio where it could be "played back" on the other Synclavier.
In terms of how each sound would have been set up and programmed on the Synclaviers, that's a very broad question that's beyond the scope of a single SE answer. You could ask about specific sounds at https://sound.stackexchange.com and get some deeper understanding there.
Note that the voice for the computers in all three 80s - 90s series were done by Majel Barrett, who was also Nurse Chapel in the original series and Lwaxana Troi in the new ones
I found a Google book with more information about the transporter sound specifically:
For example, the producers of STNG wanted to update the very recognizable sound of the Enterprise transporter to make it sound more high-tech and intense. But the producer of the original show wanted the original sound. In the end, [sound designer James] Wolvington says, Roddenberry stepped in a suggested using the original, but adding a sense of mystery. So Wolvington took the root musical chord and applied a series of tri-tones [(a musical interval)], performed on the Synclavier.
[Wolvington said,] "But I feel it would be inappropriate for me to take the title [of "Sound Designer"] because I'm making use of sounds created by Alan [Howarth], Frank [Serafin], Doug [Grindstaff], Mark [Mangini], Richard [Anderson]."
That last quote alludes to the fact that sound design often uses a combination of samples from earlier work and new synthesized sounds and samples, layering things together to get the final sound.
Effects: Cut by Jim Wolvington on a basic Synclavier setup (20 MB RAM, 32 voices), locked to 3/4 inch video. Sounds cataloged on 1.2-gigabyte magneto-optical drives, performed on the Synclavier, then laid back to 24 tracks of Tascam DA-88 [made possible by synchronizing three 8-track machines together]. Four DA-88s on the stage [I think this means the mixing sound stage where everything is put together] to pop in hard effects.
A mix of partly dubious and partly interesting information here (e.g., TNG transporter sound was a mix of synth and vocal chords, kind of like the THX sound): http://forum.vintagesynth.com/viewtopic.php?f=12&t=66843
Also, you can get your own digital recreations of the Synclavier and make your own sounds: http://www.synclavier.com/