The official name for the April 1 holiday in the United States is April Fools’ Day. “Fools” is plural because (presumably), there’s more than one fool; thus, the apostrophe comes after the “s” because of plural possession. Both The American Heritage Dictionary and Merriam-Webster’s Dictionary list the name as “April Fools’ Day.” April Fool’s Day can be considered a variant.
Interestingly, the Oxford English Dictionary lists “April Fool’s Day,” and Google’s Ngram chart suggests that the singular form (April Fool’s Day) is indeed more popular.
The AP Style Guide, The Chicago Manual, and the Government Printing Office endorse the plural form. If you are writing with a particular style guide, be sure to double-check the guide for proper usage. When in doubt, go with “April Fools’ Day,” since it’s the official name.