Push the envelope
To attempt to extend the current limits of performance. To innovate, or go beyond commonly accepted boundaries.
This phrase came into general use following the publication Tom Wolfe's book about the space programme - The Right Stuff, 1979:
"One of the phrases that kept running through the conversation was ‘pushing the outside of the envelope’... [That] seemed to be the great challenge and satisfaction of flight test."
Wolfe didn't originate the term, although it's appropriate that he used it in a technical and engineering context, as it was first used in the field of mathematics.
The envelope here isn't the container for letters, but the mathematical envelope, which is defined as 'the locus of the ultimate intersections of consecutive curves'. In a two-dimensional example, the set of lines described by the various positions of a ladder sliding down a wall forms an envelope - in this case an arc, gently curving away from the intersection of the wall and floor. Inside that envelope you will be hit by the ladder; outside you won't.
(Note for the mathematically inclined: it might seem intuitive that the centre point of the ladder would follow that same arc. In fact it describes a circle centred around the origin).
That's enough mathematics. The point is that an envelope is that which envelops. The phrase has something in common with an earlier one - 'beyond the pale'. Inside the pale you were safe; outside, at risk.
In aviation and aeronautics the term 'flight envelope' had been in use since WWII, as here from the Journal of the Royal Aeronautical Society, 1944:
"The best known of the envelope cases is the 'flight envelope', which is in general use in this country and in the United States... The ‘flight envelope’ covers all probable conditions of symmetrical manoeuvring flight."
That envelope is the description of the upper and lower limits of the various factors that it is safe to fly at, that is, speed, engine power, manoeuvrability, wind speed, altitude etc. By 'pushing the envelope', that is, testing those limits, test pilots were able to determine just how far it was safe to go. By 1978 the phrase was in use in print. In July that year, Aviation Week & Space Technology magazine had:
"The aircraft's altitude envelope must be expanded to permit a ferry flight across the nation. NASA pilots were to push the envelope to 10,000 ft."
The following year, Wolfe picked up the phrase and it went from a piece of specialist technical jargon into the general language.