Last Thursdayism is an idealized, and satirical, reformation of a number of related arguments, the most well-known of which is the young Earth creationism argument that the Earth was created 6000 years ago and that any fossils, stars, or mammoth bones that might indicate otherwise were put there by supernatural beings (God or the devil) to fool us.
As the name suggests, the central and only tenet of Last Thursdayism is that the universe was created in its entirety last Thursday, complete with false memories of last week, newspapers from last month, and books falsely claiming to be published in 1895. There is no way to disprove this theory -- it is theoretically possible. Likewise there is no way to prove the theory -- such an aetiology would leave no testable evidence.
This idea is by no means new. It is not substantially different from Zhuang Zhou's 2000-year-old observation:
"Once upon a time, I, Zhuang Zhou, dreamt I was a butterfly, fluttering hither and thither, to all intents and purposes a butterfly. I was conscious only of my happiness as a butterfly, unaware that I was Chou. Soon I awaked, and there I was, veritably myself again. Now I do not know whether I was then a man dreaming I was a butterfly, or whether I am now a butterfly, dreaming I am a man. There must be some distinction between Zhuang Zhou and the butterfly."
Bertrand Russell expressed the idea more exactly in 1921, stating that "[t]here is no logical impossibility in the hypothesis that the world sprang into being five minutes ago, exactly as it then was, with a population that 'remembered' a wholly unreal past. There is no logically necessary connection between events at different times; therefore nothing that is happening now or will happen in the future can disprove the hypothesis that the world began five minutes ago." (The Analysis of Mind, 1921)
This is not so much an argument, but rather an interesting statement of strong skepticism. It does, however, illustrate the practical futility of presuming that we should dismiss natural evidence in support of a favored theory; once we start claiming that large parts of natural history are fake, we can claim that the entire universe is fake using exactly the same argument.
Last Thursdayism is a modern coinage; Bertrand's 'five-minute hypothesis' has been around since the 1920s, while the more religious term 'Omphalos' was coined in 1857 by Philip Henry Gosse. The first known use of the term 'Last Thursdayism' was on November 2, 1992, in a Usenet post entitled Last Thursdayism proven!:
"As everyone knows, it was predicted that the world would end last Wednesday at 10:00 PST. Since there appears to be a world in existence now, the entire universe must therefore have been recreated, complete with an apparent "history", last *Thursday*."
The terminology stuck, but currently has some competition; a competing theory has it that the world was created last Tuesday, but that unlike Last Thursdayism, Last Tuesdayism holds this happens every Tuesday.