What are the UniFi Cloud Keys?

Essentially it is a compact hardware platform that runs the UniFi Controller software. 

What do they do?

the Cloud Key range is designed for compact, low maintenance deployment of the UniFi Controller server software, ideal for use in SoHo, small to medium hospitality and residential spaces, where scaling the number of clients and access points is limited in the long term and hardware costs need to be kept down.

What are the advantages of using one?  

It is small (about the size of a USB power bank), can be powered directly from POE, requires little maintenance, has removable storage for backup files in case of hardware/software failure and the newer versions (UCK-Gen2 / Gen2+) feature safe shutdown to avoid software corruption in the event of a power failure.

Other things to consider

The Cloud Key does have its limitations, primary amongst them is that the hardware is not very powerful.  Supporting limited adopted devices and clients (recommended maximum is 1000 Clients /  23 access points for the UC-CK and 2000 clients / 40 access points for the UCK-Gen2 and Gen2-Plus).

Finally, the UCK-Gen2-Plus includes the UniFi Protect software and separate 1TB storage drive, which allows for surveillance using the UVC range of cameras. The disk drive can also be swapped out for higher capacity - up to 5TB for increased footage storage.