While the phrase "power cycle" appears to be a noun, it is actually more commonly used as a verb. In simple terms, to power cycle a device means to turn it off and turn it back on again. For example, the user manual of a router may ask you to power cycle the router if it stops responding. This may mean switching the power to OFF and then ON again or may require physically unplugging the device and then plugging it back in again. Power cycling is often synonymous with resetting a device.
As we all know, computer equipment can be rather finicky at times. A device that was working fine ten minutes ago may begin acting strangely or may not be responding at all. Often the low-tech solution of simply turning off the device and turning it back on again will fix the problem. This is because information stored in the device's RAM may have gotten corrupted and caused the device to hang up or stall on a certain instruction. Power cycling the device erases the RAM and allows it to boot up with fresh information. Typically it is a good idea to wait 5 to 10 seconds before turning the device back on to make sure it has chance to fully reset. Of course, if you need to power cycle your computer, you should save any work you currently have open, since it will be erased from the RAM once the system is restarted.