Rigid or mechanical adherence to bureaucratic rules and regulations especially those involving unnecessary paperwork.
Legal and official documents have been bound with red tape since the 17th century and continue to be so. The first reference I can find to this practice is the 1696-1715 Maryland Laws:
"The Map upon the Backside thereof sealed with his Excellency's Seal at Arms on a Red Cross with Red Tape."
We now usually mean fussy or unnecessary bureaucracy when we refer to 'red tape'. The first record I have of it being used in that sense is from The pleader's guide, 1796. This spoof verse, purporting to be the work of John Surrebutter (a deceased barrister) was a satire on the fussiness of English law. It includes the lines:
Nor would the Fates... Cut the red-tape of thy years.
This is part-way towards a metaphorical usage of the term, albeit still clearly referring to actual lawyer's red-tape. The first entirely figurative usage of 'red-tape' that I can find is in Edward Bulwer-Lytton in Alice, or the Mysteries, 1838:
"The men of more dazzling genius began to sneer at the red-tape minister as a mere official manager of details."