Have you every wondered, if the duration of the day changed with time, how difficult it would have been to keep track of time with a watch which we all use! Fortunately the duration of the day is constant and any further subdivision is not tagged to an astronomical phenomenon; like the way it is done with year (tagging it to days which is marked by revolution of earth, keeping a balance of .2425 to accumulate each year). That is why no adjustment is necessary to keep the watch synchronized with actual duration of the day. Thus if a watch with enough power and accuracy is set inside a deep cave which does not have any human access and the watch is retrieved after thousands of years; it should show the correct time (as synchronized with the part of the astronomical day). Any variation to the length of the day (even by fraction of seconds) can have impact on the time of the watch! Thus any astronomical event which has impact on the duration of a day is important to note, so that the watch is synchronized.
Now what should be the best way to sub-divide the day so that we can keep track of smaller measures? Well! various civilizations have various answers to this question. Since the subdivision is not related to any astronomical phenomenon, any such division will be just a matter of convention and ease of use. The most common sub-division is dividing the day into 24 hours. Now why 24 and not 30 and not 60? Well to understand this we need to go to the beginning of civilizations where it was first devised and followed. I am not sure about the rest of the world, but in Jyotisha Shastra it corresponds to 24 horas. Hora is derived from Ahoratra (A+HORA+TRA), meaning day and night and thus signifying the influence of 7 planets on human affairs.
Now anyone can ask, why do we need to link them to planets. I would say this is for the same reason, why the days are named after the planets! The convention which developed naturally (with development of human consciousness) with time should have some merit! However, there is no further sub-divisions of Hora in Jyotisha, showing the lack of necessity of breaking time even further when it comes to human affairs. However there is another measure which is used for recording the birth time so that the horoscope can be casted properly. This is done by dividing the day into 60 ghatikas. Since the purpose is just to measure the duration elapsed between the time of birth and that of Sunrise or Sunset, there is no lordship assigned to them. Each of them is further sub-divided into 60 Vighatikas.
Vighatika is the smallest measure of time, which is enough for the purpose of casting a horoscope till the smallest varga (known so far). Lets see which is the smallest varga which can be calculated with the knowledge of correct Ghatika and Vighatika. We know that in the Rasi Chart, 12 lagna rises in a day which makes the duration of each rasi as 5 ghatikas or 300 vighatikas. This shows that the rasi can be divided into 300 parts to find the exact position of the lagna and this varga is called the ardha-nadiamsa. Based on the principles of Karma, it explains why a person would be born in that exact moment and not a vighatika before or after.
The current practice of measuring time within a day is a combination of Horas and Vighatikas. There should be a good reason for the measure to evolve this way or may be this is due to influence of multiple cultures on the place where it evolved.
The formula for conversion between these measures are:
1 day = 24 hours = 60 ghatika = 3600 vighatikas
1 hour = 60 minutes = 2.5 ghatika = 150 vighatikas
1 minute = 60 seconds = 2.5 vighatikas
1 vighatika = 24 seconds
You might be wondering what is the meaning of the term Ghatika and Vighatika. The name came from the instrument used for measurement of the these ghatikas and vighatikas, which is a Ghatika Yantra. This is made of a Pitcher with a hole in the bottom and kept inside a pool of water. Ghatika was measured in terms of the duration for the pitcher to fill up and sink! What an innovative way of measuring time :-).