The victim of a trick played on April 1st, or the trick itself.
March 25th used to be New Year's Day in England; April 1st marked the climax of the new year's revels, when tricks were played. There are other possible explanations of the source and the origin is uncertain.
William Congreve first recorded the term, in The Old Batchelour, 1693:
"That's one of Love's April-fools, is always upon some errand that's to no purpose."
So, why not some other month, e.g. 'September fool'? April has often been associated with newness and a sense of innocent naivety. Newness is straightforward - April is in Spring, the start of the year. Shakespeare alludes to the sense of naivety in As You Like It, 1600:
[Rosalind] "...men are April when they woo, December when they wed"
That is, men behave childishly and naively when first in love. The name 'April esquires' also refers to newly made squires. This goes back to 1592, as seen in Roger Greene's Upstairs Courtier:
"Two pert april esquires; the one had a murrey cloth gowne on."
April seems to have the right associations for the kind of innocent prank that requires a rather gullible and inexperienced target.
April 1st was called All Fools' Day until the 19th century and is first recorded as such by Jonathan Swift in 1712 (quoted in Hone's Every Day Book - 1826):
"A due donation for All Fool's Day."
The name April Fools' Day isn't recorded until much later. The first example I've found is a citation in the 1903 Encyclopedia Britannica.
See other phrases and sayings from Shakespeare.