It means that from the starting point in the late fifties, the computational power, whatever it was its figure at that point, has doubled approximately 30 times since then, meaning that, if it was, say, 10 at the beginning, then it became 20, then 40 , then 80, then 160, and so on for thirty times.

The correct phrase is "something IN the order of". It means "about".

"On the order of" is a physicist's or mathematician's way of saying "about the same--to within a factor of ten or so." "30 doublings" would be multiplying the computational power by 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 times, or 2^30 times, or 1,073,741,824 times. "On the order of 30 doublings" means "you can't really measure it all that precisely, but computation power has increased by a factor of maybe a billion." 3 billion would still be "on the order of a billion," 400 million would still be "on the order of a billion." Not 100 million, not 10 billion, but on the order of a billion.

You're welcome.

'In the order of' means 'in the order of magnitude of', that is 'in the size of'.