The regions are relatively easy. Names such as Bretagne (Brittany), Normandie, Alsace and Lorraine are familiar to anybody who knows anything about France and even the less well known ones are big enough and have obvious enough names that you can pick them up fairly quickly.
The départements are another story. They're much smaller than the regions, and there are 95 of them in metropolitan France, each with a name and corresponding number. Many of the names come from rivers and, as a result, are harder to remember because rivers are long and thin and flow from one place to another. The Centre region, for example, is divided into 6 departments: Eure-et-Loir, Loiret, Loir-et-Cher, Cher, Indre and Indre-et-Loire (and if anybody knows why there are two spellings for Loire, please post a comment!).
The numbers mostly correspond to the name of the department's position on an alphabetical list, so that Ain is number 01 (not to be confused with Aisne, which is number 02) and Yonne is 89. There are, however, some exceptions which provide interesting trivia for the true département geek. Corsica used to be one department, number 20, but was split in half and the 20 became 2A and 2B (not to be confused with 02 above), so there is no number 20. Likewise, departments 91 to 95, which lie around the outskirts of Paris, are newer and therefore not in order. The Yvelines was created at the same time but has number 78, which was the number of the old Seine-et-Oise that, along with some of the other new departments, it used to be part of.
So why is learning this fabulous list so important to the residents of France and how do they manage it? Well firstly, the department number forms the first two digits of each postcode, so unless you're like me and have lived in so many places that you're at risk of forgetting your current address, never mind the one you lived at 5 years ago, everybody knows their own and that of their nearest and dearest (or at least the people they write letters to).
Secondly, and far more interestingly, however, the number also appears on car registration plates. It used to be part of the registration number itself but on modern plates it's at one side. Its presence means that when somebody rudely cuts up a farmer driving his tractor along a road in Brittany, rather than merely jumping to the conclusion that such a person must be a Parisian, the Breton can have his suspicions confirmed by the 75 number plate. Likewise, when Parisians are held up by someone with a distant department number crawling round the périphérique, they can enjoy the satisfaction of knowing that they are superior to this rural cousin with no knowledge of the great metropolis.
Understanding Frenchman, who always likes to look on the bright side, claims that Parisian drivers are more tolerant of those with "foreign" numberplates than they are of fellow franciliens. Which kind of makes me wish I'd been able to take my car somewhere far, far away to be registered!