In 1793, the French switched to French Revolutionary Time, creating a decimal system of time. A day had 10 hours, 100 minutes per hour, and 100 seconds per minute. The system was elegant, doing away with the complex math required for time calculations under a 24 hour/60 minute/60 second system. But it also brought huge headaches.
French Revolutionary Time came alongside the French Republican Calendar, a further attempt to rationalize time. Months were divided into three 10-day weeks, and there were 12 months. The leftover days needed to add up to 365 or 366 for the year were tacked onto the end of the year as holidays. This was a bit inelegant (days and years being hard to divide cleanly by 10), but at least it was less confusing than trying to sort out what time "noon" was (it was 5 o'clock).
French Revolutionary Time only lasted 17 months. By April 7, 1795 (in the Gregorian calendar), the time system became optional. Decimal clocks and decimal/standard hybrid clocks continued to be used for years, but for practicality, France returned to the same system of time as its neighbors.
The French Republican Calendar lasted far longer. It began in late 1793 and ran all the way through the end of 1805 (again in the Gregorian reckoning). On December 31, 1805, the French government chucked the system—in the year XIV, by Republican reckoning. This was due, of course, to the reign of Napoléon Bonaparte as Emperor. (Incidentally, his coronation occurred on 11 Frimaire, Year XIII of the French Republican Calendar—also known as 2 December, 1804. It took him more than a year to roll back the revolutionary calendar.) In any case, January 1, 1806 rolled around using the Gregorian calendar and the rest is history.
Of course, all this calendar-nerd stuff leads to the fact that you could still choose to use the French Republican Calendar. Indeed, Wikipedia will tell you the current day and year using the system, although you'll want to read up on the exquisite problems related to leap years (also helpfully detailed on Wikipedia).
For a bit more on decimal time (including several modern variants), check out our article Decimal Time: How the French Made a 10-Hour Day.