The eurorack standard is one among many modular synthesis standards. This typically contains things like module size (eurorack uses 3U or 133,35mm module height) and the powersupply voltage (eurorack uses ±12V and sometimes an additional 5V). The width of modules is meassured in HP ("Horizontal Pitch" or TE "Teilungseinheit" in German). One HP equals 1/5 inch or 5,08 mm. For further Details on these Standards have a look on Doepfer's Instructions Page.
A eurorack synthesizer consists of one or more modules, which are patched together via mono 3.5mm patchcables (this is the mono version of what the typical headphone plug looks like). The signals deviate between ±10 volts but can be even bigger in some cases. This is why it is not advised to plug a eurorack signal directly into other audio hardware unless it is e.g. a mixer with enough headroom or a soundinterface which can lower the levels with an input pad.
The awesomeness of modular synthesis is its modularity. With only a few modules patched into each other and into themselves a unpredictable flow of sounds and signals can be created. If one module reveals to be useless it can be usually sold without much loss and replaced with another. There are modules which can emulate a much more expensive analogue drum machine (A Roland 808 could be emulated with the modules from Tiptop Audio for example), there are modules which help you control the modular synthesizer from your Laptop (like the ES-3 by Expert Sleepers) or the other way around. There are setups that are meant to be triggered and controlled live like with a keyboard, a ribbon or theremin controller, there are setups that basically automatically play forever evolving sequences, without human guidance needed, there are setups which just use the eurorack as a fancy effects machine. It is not always about fully understanding what is going on, but about listening to what happens and responding to it.
A powersupply which delivers ±12V and more current than all modules together are drawing
A busboard which helps distributing the ±12V to more than one module.
A case with rails to mount the modules and the powersupply
All of these are available commercially or can be done yourself. In the Eurorack world DIY can often be cheaper than buying commercially available products – if you already have some parts and tools needed.
Where to get
A good place to explore different modules is modular grid. Here you also can create and plan a virtual rack and it will tell you the overall power consumption.