Why a day is divided by 12/24 hours? Why the number 12?

12 h divides into many whole numbers and so it is easy to think about portions of a day:

• A half = 6 h
• A third = 4 h
• A quarter = 3 h
• A sixth = 2 h.

You can't do this with 10 h, 6 h, 14 h, or 16 h as easily.

I'm guessing they started with inscribing a regular hexagon in a circle. Next draw the three diagonals, which are also diameters of the circle, and construct for each the perpendicular line through the center. Voila, you've divided the circle into 12 equal sectors. Alternatively, constructing an inscribed regular dodecagon in a circle isn't all that hard either:

By this rationale, 10 = 2*5 wouldn't have been a likely choice because the method of construction of a regular pentagon wasn't discovered until the time of Euclid, and 14 = 2*7 would be most inconvenient because the regular heptagon isn't constructable with compass and straightedge.

Image Source: http://upload.wikimedia.org/wikipedia/commons/2/2b/Regular_Dodecagon_Inscribed_in_a_Circle.gif

12 was a round number back then.

The dozen is relatively easy for otherwise uneducated people to learn to count to on one hand. It was the base for a few ancient numbering systems, instead of the now-ubiquitous base ten.

https://en.wikipedia.org/wiki/Duodecimal

"The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers). It is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5. In this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full."