In 1965, Intel co-founder Gordon Moore wrote an article called "Cramming more components onto integrated circuits."
In the article he said
The complexity for minimum component costs has increased at a rate of roughly a factor of two per year. Certainly over the short term this rate can be expected to continue, if not to increase. Over the longer term, the rate of increase is a bit more uncertain, although there is no reason to believe it will not remain nearly constant for at least 10 years. That means by 1975, the number of components per integrated circuit for minimum cost will be 65,000.Note that he is not talking about what would be the largest theoretically possible chips, but about what would be cost effective.
Moore's prediction was based upon extrapolation of the history of the integrated circuits up to that time:
Note that he is predicting exponential growth -- growth at a constant percentage rate.
He does not use the term "Moore's Law" in the article, but the term/meme caught on and we are still using it to describe exponential growth of all things techie -- storage and memory density and speed, communication speed, etc.
Moore's projection held up well beyond 10 years. In this plot of the number of transistors on commercial CPU chips through 2011, the line represents doubling every two years:
The accuracy of his projection is all the more remarkable when you realize that the prediction was made six years before Intel's first CPU chip, the 4004, which had 2,300 transistors.
At some point, density increases will level off, but that point has not yet been reached. Apple's 8X system on a chip that is inside your iPad Air has 3 billion transistors.
The first electromechanical compputers used electromagnetic relays as switching elements. Folloiwing genertions moved to vacuumn tubes, transistors and today's integrated circuits. When Moore's law finaly hits the wall, will we move to another switching technology and continue improvement?
You can check out some cool Moore's Law infographics here.
A Re/code article says Moore's law is 50, but may not reach 60. The article quotes Intel executive Tracy Smith as saying they expect to be making chips with 5 nanometer features (about twice the size of a strand of DNA) around 2022, but that will be the end of the line.
The article goes on to speculate on what technology might come next -- the red question mark in the above figure -- but makes no predicitions. It also includes the following video (1m 53s) of Gordon Moore reflecting back on his 1965 article and the term "Moore's law," coined by semiconductor pioneer Carver Mead.