(just a heads up, i'm gonna be working on this page for a few days)
Modular synthesizers are essentially regular synthesizers broken up into their individual sections (oscillator, filter, etc.) connected together by patch cables (typically 1/4" or 1/8", but occationally banana plugs). There are a wide range of units available in a range of formats such as Eurorack from Doepfer, and 5U from Synthesizers.com.
Modular systems are a great way to breathe new life into boring sounds - take this track for example.
module explanation[]
Modular synths can include some interesting module types, like function generators and polarizers, for example. Some of the more generic and basic units include modules like Voltage Controlled Oscillators (VCO), Voltage Controlled Amplifiers (VCA), or Voltage Controlled Filters (VCF). These all do what you'd expect them to - oscillators make noise, amplifiers amplify that noise, and filters filter that noise, they're generally just named in like, analog synth terms or something.
MIDI inputs are usually required to create audio signals based on what you create in your DAW. MIDI interfaces are usually in the form of Control Voltage (CV) converter. There are a bunch of different signals that are produced by CV interface modules, such as gate signals, trigger signals, and pitch signals. A quick google search should explain the differences between the 3, it isn't really that complicated.
You may also hear about modules called multiples, which essentially copy the audio signal and control voltages. There are two types of multiples - passive and buffered. Passive multiples don't use electricity, and are sockets which are all connected to each other, meaning the audio signal going into the input will be produced in all sockets of the multiple. There are a bunch of different types of passive multiples and there are plenty of videos on youtube that go into more depth. Buffered multiples use electricity to create exact copies of a sound, and have a few dedicated inputs and outputs so that more than one audio signal can be run into the multiple and more than one signal can come out of the multiple. It is quite rare to require a buffered multiple, and usually passive is all that is necessary.
not quite a buyers guide[]
As previously mentioned, there is a vast ocean of modules that span a vast range of formats. It is possible to buy pre-built modular synthesizers, such as Pittsburgh Modular's Foundation 3.1, however buying pre-built can limit your synthesizer's capabilities, make you spend more money, and is significantly less fun when compared to buying units and constructing your own customized synth - building a modular system could probably be compared to building your own PC, for instance.
As with any synthesizer, the modular system will depend on what kind of music is being created. ModularGrid is a good site to begin planning which modules you want in your build, as well as getting an idea of how a modular system works.