Long ago, Alice got tired of burning her lunch. At the same time, she noticed a kid fill a bottle with sand, then wait for the sand to run out. It took about the same time for the sand to run out of the bottle every time; perfect! Alice made a really big bottle, and put marks on the inside. One mark for "boil an egg", one mark for "grill a steak", and so on. Fill the bottle to the desired time, turn it over, and as soon as it ran out, the food was ready. Alice and her husband called it "our sand-bottle", which is where the term "hour" comes from: the time it took for the bottle to run empty at its highest setting.
Then, Bob got the great idea to use water. Water trickled over a waterfall at a fairly constant rate, so he stuck a bottle under it. When the bottle was full, it would fall over; with a small machine to reset the bottle, it was a great timekeeper, since it didn't need to be reset. The bottle could be very minute; a "minute" bottle. It took 60 minute bottles falling over to make an our-bottle: 60 minutes in an our. Even better, a stack of numbered bottles meant the first real clock had been invented.
For a long time, "ours" and "minutes" were the de facto standard. That is, until a wizard noticed that when water flowed faster, his minute bottles fell faster. He got tired of time measurements varying when he was at his lab and when he was at home, so he built a tall tower with a magical light at the top. Every time a bottle fell, a light was added, 60 lights in total. Perfect! He could easily measure minutes and ours by looking out a window. He decided these would be named "clonks", due to the noise they made when changing lights.
Days and Time Zones
Soon, the next town wanted one. So, he built another light tower, only this time, instead of hooking it up to a bottle, he set up a sensor that watched the other tower blink. That way, both towers were synchronized. There was a little bit of lag, but no one really noticed. Eventually, he realized that everyone would want a tower, so he set off on a great journey, to place clonk towers in a ring around his world. However, he realized that the lag would be an issue; to fix the problem, he decided to to set up a series of small clonk towers, then an ominously huge clonk tower, or O-clonk. Each huge tower would be one "our" apart. His friends begged him to use an even number, but he decided that 24 "ours" around the planet would be to his liking. His clonk towers were a huge success, and soon people everywhere knew if it was 5 O-Clonk. The time it took for a signal to go around the world was known as a "day", or so the wizard said. No one knows why he called it that. Wizards are weird that way.
Finally, as time went on, people needed finer measurements. Someone decided to add a second set of even smaller bottles to the first; finally, the "second" had been invented. Some wise-guy made a series of 1000 bottles that took only 1 second to tip over, but his "many-second" never really caught on.
Other units of time
Other units of time sprang up naturally: going 7 "days" without any food made one "weak", it took roughly 30 "days" to recover from "mumps", a famous comedian always some 365 "days" to come up with a new "jeer", and so on.