Futilely pursuing something that will never be attainable.
The most popular theory as to how this phrase originated is that it comes from a form of 16th century horseracing, where riders had to follow the lead rider through an erratic course. This is said to resemble flying geese that follow, or 'chase' their formation leader.
In harmony with the 16th century horseracing date, the phrase is actually used by William Shakespeare in the play Romeo and Juliet, believed to have been written in 1594 or 1595:
"Nay, if thy wits run the wild-goose chase, I have done, for thou hast more of the wild-goose in one of thy wits than, I am sure, I have in my whole five: was I with you there for the goose?"
* While looking all over town for a man named Billy, it began to feel like a wild goose chase, because I don't think I'll ever find him.
Find more sayings with this handy list of phrases. It has their meanings and other interesting information!
knowyourphrase.com - Meanings and Origins of Phrases and Idioms
Note: A phrase's origins are, much of the time, completely unclear. The origins you see listed are the plausible theories floating around for how or where an idiom came from, but may not necessarily be accurate. The quotes which have the phrase in them are the oldest written forms of the phrase I could find, but keep in mind that older recordings are probably out there somewhere. If you find an earlier citation than what I have, feel free to let me know!
Also, remember that just because you see a saying in an old book or newspaper, let's say from the year 1893, it does not mean the saying originates from that source. In all likelihood, if an expression is already in use in a book or newspaper, then it's probably older. Nevertheless, these old quotes serve as a way to show the reader how far back in history some of these sayings go.