The Martian clock and calendar are significantly different than the Earth clock and calendar, although we tried to keep things as much like home as possible. Fortunately for those that have strong body rhythms, the Martian day is nearly the same as the Earth day, so you shouldn't lose any sleep over the shift in planets. But the extra 39.5 minutes is enough different that it causes problems. If Mars had been colonized by businessmen, we would have just slowed our watches down by 3% and continued using sols with 24 hours, hours with 60 minutes, and minutes with 60 seconds. The only difference would have been that the second on Mars would have been 3% longer than the second on Earth.
Unfortunately, this is heresy to a scientist. A second is no longer 1/86,400th of an Earth day, but is defined as the duration of 9,192,631,770 cycles of a certain frequency of radio waves from a cesium atomic clock. Since Mars was colonized first by scientists, we are stuck with the system that they developed, where a Martian second is exactly equal to an Earth second, so the scientists on the two planets can talk straight to each other. You will have to put your Earth watches away and get a Mars watch.
The Martian day is called a sol. The Martian sol is divided into 24 increments called maurs (for Martian hours). Each maur is roughly the length of an Earth hour so students still have one hour class periods and people still work eight hour shifts, only now they are eight maur shifts.
Now comes the real difference. Instead of a maur being made of 60 minutes of 60 seconds each for a total of 3600 seconds per hour, there are 60 marmins (Martian minutes, of course) of either 61 or 62 seconds each (remember a Martian second is identical to an Earth second). The shorter marmins are the 00 and 01 marmins in each maur and every marmin divisible by 3, for a total of 21x61 + 39x62 = 3699 seconds per maur. Then, three sols out of four, the 02 marmin of the 00 maur has only 61 seconds to give 3698 seconds for that maur and a total of 88775 seconds for the sol. The fourth sol does not skip that leap second and is 88776 seconds long. This gives an average for the four sols of 88775.25 seconds, close enough to the physical sol that the difference over a Martian year can be included in the leap seconds the scientists add or subtract occasionally anyway to adjust for the slowdown in the planet's spin rate and changes in the shape of its orbit. Fortunately, you don't have to worry about memorizing where the leap seconds go or don't go, all the leap seconds are programmed into the timing circuit of your digital Marswatch, along with the various holisols and leap sols to keep the yearly calendar on track with the seasons.
Another major difference between timekeeping on Mars and on Earth is that, because of the greater distance of Mars from the Sun, the Martian year (called a mear) is nearly twice as long as an Earth year. Because we have no slow-moving moon to make "moonths" important, we have done away with them. But we have kept the 7-sol week. (T.G.I.F. means the same thing on Mars as it does on Earth.) A Martian calendar year consists of 95 weeks of 7 sols each for a total of 665 sols.
To round the calendar up to the nearly 669 sols of the physical mear, we add four seasonal "holisols" outside the weekly calendar as near the physical equinoxes and solstices as possible. They are: the perisolstice, the apequinox, the apsolstice, and the periequinox. The following sol starts a new week, so is always a Sunsol. Thus everybody gets two sols off in a row.
The mear is really about 668.6 sols long, so every 2.5 mears, starting from M001.25 or the apequinox of M001, we have either a "springmear" and spring forward over the periequinox into the new week, or a "fallmear" and let the apequinox fall off the calendar, so the start of each new mear stays on track with the seasons. That should keep us on track until way past M999 (about A.D. 4000) when everyone will have to buy a new watch anyway.
The 95 weeks are divided into seasons. The seasons are not equal in length because of the high eccentricity of the orbit of Mars. Since the main human habitation here on Mars is now in the southern hemisphere, we designate the seasons by what is happening here. Those that live "Up Over" have the opposite seasons, of course. The year starts with the perisolstice, the solstice that occurs when Mars is close to the sun. This starts the short, hot summer in the southern hemisphere and the short, warm winter in the northern hemisphere. This southern summer season is 22 weeks long. There follows the apequinox and a 27-week fall, the apsolstice and a 26-week winter, and then, as Mars starts back in toward the Sun, is the periequinox followed by the shortest season of all, the 20-week spring.
The seasons are not used in writing down dates. The standard order for a date is a letter M to indicate that it is a Martian date, immediately followed by the number of the mear, with M000 coinciding with the Earth year 2000, when the starting times of the Earth year and the Mars year were very close. (They weren't perfect, but let the scientists worry about those little details.) Then comes the number of the week from 00 to 95, the three letter code for the sol of the week (SUN, MON, TUE, WED, THU, FRI, SAT), the maur, the marmin, and the second. If the date happens to fall on a holisol, the five letter abbreviations PSOLS, AEQUI, ASOLS, and PEQUI, are used instead of the week and sol. In this nomenclature, the start of the mear is M025/PSOLS/00:00:00, while the last second of the mear is M025/95SAT/23:59:62.