When you're working around the clock, you're busy all day and all night long. Around the clock means for an entire 24-hour day.
The phrase around the clock is used most commonly when someone is hard at work, or is doing something difficult. You might say that a new baby seems to cry around the clock, or that the emergency room in your city is busy around the clock. It's easy to understand how this saying arose when you imagine the hour hand on a clock moving all the way around its face twice — in other words, marking a whole day.