In France, April 1 is referred to and celebrated as poisson d’avril (literally “April fish”). As I understand it, people can play any kind of prank, but the classic joke is to attach a paper fish to a victim’s back without being noticed. (Apparently this is also found in other nations, such as Italy, where the term is pesce d’aprile.) The successful jokester of any prank might declare “poisson d’avril!” as one might declare “April Fools!” in the states. I have seen suggestions on English language sites that the victim is declared a poisson d’avril, but I believe that is incorrect. The joke, not the victim, is the poisson d’avril.
There seems to be some consensus that the poisson d’avril tradition in France started following calendar reform in 1564 that moved the start of the year from the end of March to January 1. According to this site, the reforms disrupted “a traditional week-long celebration of spring and rebirth that had lasted until…April 1.” This English-language travel website explains, “in a time without trains, a reliable post system or the internet, news often traveled slow and [folks] in rural France were the last to hear of and accept the new calendar. Those who failed to keep up with the change or who stubbornly clung to the old calendar system and continued to celebrate the New Year during the week that fell between March 25th and April 1st, had jokes played on them. Pranksters would surreptitiously stick paper fish to their backs.” This article has a similar explanation, although it suggests the calendar reform and April fooling were not limited to France.
This French site has a somewhat different (and seemingly more authoritative) take. It also focuses on the 1564 calendar reform, but — if I’m understanding the French correctly — explains that, because people were used to end of year gift exchanging at the end of March, folks continued to exchange small gifts at that time, perhaps to cast doubt on or make fun of the calendar switch. Over time, the gifts became pranks.
The site also posits an explanation for the fish. It explains that the April gifts were often food. Because April 1 often falls during Lent, during which the consumption of meat is forbidden, fish was a common gift. Thus, a common prank was an offer of a fake fish. The online Encyclopaedia Britannica has a different explanation: “the fooled person is called poisson d’avril (‘April fish’), perhaps in reference to a young fish and hence to one that is easily caught.” A final theory is that the fish is in reference to Pisces (Poissons), the last Zodiac sign of winter, ending in late March (although April 1 is under Aries).
Here is the French Wikipedia page on the topic. There are many other webpages that address the topic, although those that I viewed simply repeat pieces of the information above.
Ultimately, my main interest is in sharing some of the amusing vintage poisson d’avril postcards I’ve collected, mostly from the period 1900-1910. The cards feature images of fish held by young men and women, reflecting that sweethearts often exchanged the cards. Interestingly, most of these cards have nothing to do with pranks and just use the fish as a general symbol of the date, and in many cards as an actual gift. Somewhat surprisingly, I’ve never seen a vintage postcard referring to the custom of sticking paper fish on a victim’s back.
Wonderings 
Fun fun fun!