A No Slot Clock is exactly what the name implied -- a computer clock that does not require an expansion slot to use. The term is generally applied to the Apple II series.
The big deal is that the first Apple II machine (previous to the Apple IIgs) didn't have a RTC. You could get card-based solutions to this problem, most well-known being ThunderClock card, but this could be a problem -- the well-versed Apple II user probably has a multitude of other cards installed and consequently had no space for another card (assume serial card, parallel card, memory card, 80-column card, modem card mouse card and a SCSI card). And, of course, let's not forget machines that didn't have slots.
The way around this was to make a clock that didn't use a slot. The no slot clocks did this by being passthru ICs that would plug in under one of the chips on the motherboard. During boot up, the code in the clock chip would make the computer think it was a clock card (usually emulating the Thunderclock), and all was well with the world.
The Apple II+ didn't have the right ROMs socketed, so you'd need a clock card. On the Apple //e it was installed under the CD (Character Device) ROM, and on the Apple IIc and Apple IIc+ it is under the Monitor ROM.
A second variant of the No Slot Clock theme were clocks that used the DIN-8 serial port on the Apple //c. These were external bricks with passthroughs so you didn't actually lose use of your serial port.