A phone jack is a type of electrical connector originally used to connect phone calls. The female part set into the hardware is the jack or socket; the corresponding male part at the end of a cable is the plug. Although phone jacks were invented for telephone switchboards, they have since become popular for all kinds of audio applications.
The oldest phone jack size still in use, introduced in the 1870s, is 6.35 mm (¼"). Since then, the evolution from vacuum tubes to transistors allowed electronics to be miniaturised. By the 1950s, the ability to make a small, portable transistor radio necessitated a smaller 3.5 mm jack.
Physical size aside, these jacks are completely functionally equivalent. It's perfectly possible to get cables with a 6.35 mm plug at one end and a 3.5 mm plug at the other end, as well as adapters that allow a 3.5 mm plug to plug into a 3.5 mm socket and vice versa.
Phone jacks of both sizes have two main types: some have just a tip and sleeve (TS), for an unbalanced mono signal; others have a tip, ring, and sleeve (TRS), for either a balanced mono signal, or more likely, an unbalanced stereo signal.
Like many other electric and electronic musical instruments, hardwired synthesisers usually route their sound to one (mono) or two (stereo) 6.35 mm TS jacks at the back.
Modern 5U Moog-style modules consistently use 6.35 mm TS jacks for all their patch cables too, for audio, CV, and gate signals alike (though Moog used a different kind of cable for S-trig in their original 900 series modular synthesisers). Similarly, 3U Eurorack modules consistently use 3.5 mm TS jacks for everything.
Modular synthesisers tend not to have a dedicated output — you can safely connect pretty much any module's output directly to a multitrack recorder, mixing desk, or sampler. I tend to use the VCA's output for this, which I believe is common practice.
Various different signal strengths are commonly used. Microphones and electric guitar/piano/organ pickups convert physical movement into an electric signal, so both inherently generate very weak signals, up to about -0.1 V to 0.1 V. These need amplification. Pretty much everything else uses line level audio signals, although professional and consumer devices disagree on what level that is exactly. Either way, it's weaker than -2 V to +2 V. CV and gate signals are a comparatively quite strong -5 to +5 V and 0 to 5 V respectively, sometimes even more.
Phone cables were originally housed in a braided cotton jacket. These days, plastics such as PVC, PUR, and TPE are by far the most common jacket materials. Even braided jackets are often made out of nylon, a sneakier plastic, although that's still a big improvement over the others. If you can afford them, I recommend cables with a braided jacket, as they tend to form a gentle curve rather than retaining a scrunched-up zigzag of kinks, making them more aesthetically pleasing both visually and to the touch.
People often point out how modular synthesisers look like telephone switchboards, and they're absolutely right. They use essentially the same technology to accomplish the same goal, only instead of patching a caller in London through to someone in New York, you're patching an oscillator through to a filter. Either way, much like Kraftwerk, you're the operator.
Signal routing: Phone jack | Pin matrix