Clicky refers to a keyswitch with a dedicated clicker mechanism. A clicky keyboard is one which uses clicky switches.
Clicky switches omit a click sound when depressed. This sound is distinct from the clack sound when a key is bottomed out. Clicky switches typically actuate at mid-travel, which means that depressing a click switch slowly will trigger a click sound before the key is fully depressed.
Some switches will inherently click when released. Many will only click when depressed, and make no additional sound when released. Some switch types will always click when depressed, and sometimes click when released, with this behaviour varying from key to key across the keyboard.
Various approaches have been taken to accomplish a click; most methods do not guarantee that the switch will click precisely when it actuates. One switch which does make this guarantee is the buckling spring, as a single change within the spring triggers the actuation sound and generates the click.