The process of transforming the electrical current of an audio signal to digital, very simply put, is like scanning the incoming sound waves, and creating a digital representation of it (imaging it transformed in a series of 0 and 1), to be then re-transformed in electrical current and then back to the physical world through your speakers again.
This is what, very broadly, happens within an audio interface.
When the electrical current is transformed into digital, it passes through the analog to digital converter of the audio interface (A/D). Here the interface takes a series of snapshots of the sound wave to be then coded in digital language. The number of snapshot taken per second is actually pretty high, and the standard minimum required by the modern industry for CD replication, for example, is 44100 samples per second.
Each snapshot, or sample, is then a value, and every sample is coded next to each other following the order in which these snapshots were taken, generating the digital representation of the electric waveform.
It is then important that the audio interface scans the incoming audio with extreme precision, otherwise, when the digital waveform is created, it might have discrepancies and not be exactly the right waveform, generating audio artefacts such as clicks, pops or sometimes your system won't even play the audio back.
This is what the infamous clock is for.
A clock, or word clock, simply generates an impulse every so often for the device to follow in order to keep the sample snapshot and conversion consistent, generating a digital representation of the waveform as close to the electric signal as possible.
When working with a single audio interface, there shouldn't be issues of clocking, as the device itself follow its own clock. However, when using multiple devices at the same time, a high quality word clock becomes fundamental.
If the devices were to follow each their own clocks, you would end up with lots of data errors, since there would be discrepancies in the digital audio generated by each device with each other. This is why it is important to nominate a master clock, or clock source, which is followed by all the devices connected to the system.
Yet, if the clock device is not sophisticated enough, there could still be discrepancies.
For this reason it is always recommended to have a specialised device to be plugged in the system and just manage the clocking. This device is called master clock device.
These devices are usually expensive and you can often get away with it just setting your best quality audio interface internal clocking as the clock source.
The master clock device will then provide the clock source to each audio interfaces connected to the system.
There are different ways to connect these devices and, even though it would be best to have them directly linked via a BNC cable, RCA or digital cables such as AES, S/PDIF or ADAT, in the most simple occasions you can just carry the clocking signal via the cable by which the audio interface is connected to your computer (usually USB, Thunderbolt or Fireware).