The Clock is a Watchdog whose name is Tock. His job is to make sure that nobody wastes time. The Lethargarians consider him a very unpleasant character because all they do is waste time. In the middle of his body is a large alarm clock that goes off when people are wasting time. When he first meets Milo, Milo tells him that he is just "killing time" (pg 30) That makes Tock so upset that his alarm goes off. He says "It's bad enough wasting time without killing it!"(pg 30) Killing time is an idiom we use in English to mean wasting time. Later he explains to Milo that once there was no time at all and people were missing trains and eating at wrong times. So they invented time, but there was so much of it (60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, 365 days in a year) that people did not value it. It there was so much of it, it couldn't be valuable. People started to waste it and give it away. He and his family have been given the noble cause of making sure that people do not waste time because time is our "most valuable possession more precious than diamonds" (pg 34) It marches on and doesn't wait for any man. Later, as they are driving along, Tock continues to explain the importance of time to Milo quoting philosophers and poets. Norton Juster, the author, does not go into detail about it, but you can tell that Tock feels he has a very important job, something that even philosophers and poets have found it necessary to discuss.
The page numbers I have given are for my version of the book. Basically this answer is in Chapter Three: Welcome to Dictionapolis.