It may help to read up on the history of human timekeeping, and the various different problems it was designed to solve.
The first and most obvious unit of time is the day. Evolution has invented the circadian rhythm to alter our biological behaviors around the day/night cycle, including a period of nighttime sleep. Melatonin is the human sleep hormone and its production is supressed by the visible presence of 400nm sky blue light, this is hardwired into our biology by evolution.
An exoplanet might also have a very short day of say under an hour, or it might be a moon with month long day/night cycle. If your planet is tidally locked, then you may not have a day-night cycle, or you may have a polar region that experiences seasonal "days" where the sun doesn't rise or set for 6 months at a time.
There was an experiment where humans were kept in a cave without access to clocks or sunlight. The humans eventually adjusted to a 36-hour sleep/wake schedule.
In a prison environment, you would simply start counting the day-night cycles from the most significant event in the history of your environment: such as upon first arriving, or a major change in social structure. Most calendars count from the birth of their founding prophet/leader.
Not all languages have the ability to count, and the linguistic framework may be limited to: one, two, three, many
The next evolution in time is recognizing other seasonal patterns, such as the moon, seasons and the year. The lunar calendar doesn't require any technology to keep track off, but it doesn't synchronize exactly with the year cycle (12.37 moons per year).
It took quite a few attempts to get the measurement of a year correct. The Roman Julian Calendar has at times been 304, 355, 365, 377 days long, with between 10 and 12 months. The usual historical design pattern is that the calendar got noticeably out of sync with the seasons, then the calendar was monkeypatched with leap months, inventing new months, or occasional redesigns to tweak the number of days in the year. Measuring a year requires statisticians to measure the seasons (such as the first blossom of spring). We are still adding leap days and leap seconds.
The year is important mostly to measure the seasons, upon which our survival through agriculture is dependant upon. It is important to know when to plant the crops and when to harvest.
Festivals are often held to mark the transition between seasons, and the changing of cultural rules/expectations. In ancient Greece, the Olympics was held every four years and marked a period during which war could not be conducted. Other festivals mark days remember historical events or times where exceptions to the usual social rules can be practice (Venice masked ball, or 29th February when a woman can propose to a man)
For a tidally locked planet without seasons, the sun is always at midday and the year might be a fairly meaningless concept.
The clock was first invented so monks could have an alarmclock for morning prayers. Before that, there was either the sundial (based on the relative position of the sun) or the hourglass (which is somewhat inaccurate). The quest for ever more accurate clocks turned out to be the solution for naval navigation and later GPS.
The modern physics method of the atomic clock is to measure the constant vibrational frequency of cesium atoms. The software complications of describing leap seconds is now driving a trend towards absolute atomic time, with a greater acceptance of allowing the solar day to become a few seconds out of synchronization with atomic time.
Timezones were only really required after the invention of the telegraph and the telephone. This beaks the sundial that assumes the shared position of the midday sun. The requirement is to synchronize clocks, even at the abuse of the solar days. Some extreme timezones can (like Western China), can be 2+ hours out of sync from the solar day. Here the timezone has almost been imposed by conquest.
An alien civilization could be forced to live under an imposed clock or timezone. Else for a colony on Mars, they would keep two different clocks and calendars: Mars time and Earth time. Different tasks and events would be aligned with each calendar, so some Mars individuals may still choose to keep an Earth-clock sleep schedule.
The Martian day (referred to as “sol”) is approximately 40 minutes longer than a day on Earth. This leads to interesting synchronization issues, equivalent to the jet lag caused on a merchant ship traveling one timezone every day.
The human body can also be trained into alterative sleep cycles such as the Uberman, which uses 6 x 20 minute naps during the day, without a 6h+ deep sleep. This is sometimes used by solo yachters who need to keep constant watch.
For an alien planet, ask the following questions:
- What are the easily observable repeatable natural events in the environment?
- Are there any important seasonal trends in the environment?
- How does behavior change during different time periods?
- What technology is available for measuring time?
- What are the major historical events in your civilization?