SPQR was one of the main terms used by Romans to describe their empire. It appeared frequently on monuments and coins from around 80 BC onward. Before then the term 'Roma' was generally used in that context. Exactly why SPQR became the norm is not entirely clear. It first appears in political speeches shortly after Sulla revived the office of dictator, so it may have become popular as a reminder of what the state was supposed to be, in a period when warlords like Sulla were prone to turning their legions toward Rome and ruling by force. We'll never know for sure, however.

After the transition to autocratic rule under Augustus, SPQR continued to be used. This was partly because the early emperors were not truly autocrats - their power ultimately came from senatorial legislation which granted them their powers - and it was good to keep up appearances. Augustus in particular wanted people to believe that he was taking Rome back into a golden age, putting Rome back the way it was before all the civil wars, so keeping the label of SPQR was a useful propaganda tool. It was also just a convenient shorthand for 'the Romans state', so continued to be used long after autocratic rule was well established.

Here is the full phrase on the Arch of Titus, in Rome: https://ka-perseus-images.s3.amazonaws.com/11d451d68c17a53d77b32d5f381725b723a52f5e.jpg

And the abbreviation on the Arch of Septimus Severus (bottom right, at the end of the inscription): https://www.ancient.eu/uploads/images/1004.jpg?v=1485680372

The term SPQR would also be used by many mutinous armies, to show that even if they were not loyal to the emperor, they were still loyal to the Roman state. On 1st January 69 AD, two legions refused to take the annual oath of loyalty to the emperor, Galba, and instead swore allegiance to 'Senatus Populusque Romanus' - SPQR. This move precipitated Galba's downfall, as it showed how little authority he commanded and that many did not consider Galba to be a legitimate ruler, hence they made a pointed distinction between Galba and SPQR. A lot of coins minted by the usurpers of 69 AD have SPQR on them, to try and distance themselves from their dubious path to power.

Some coins with SPQR on them from 69:

http://davy.potdevin.free.fr/2%20Roman%20coins%20I%20Augustus%20to%20Vitellius/7-civil%20wars/ric71.jpg http://davy.potdevin.free.fr/2%20Roman%20coins%20I%20Augustus%20to%20Vitellius/7-civil%20wars/ric65.jpg http://davy.potdevin.free.fr/2%20Roman%20coins%20I%20Augustus%20to%20Vitellius/7-civil%20wars/ric58_2000chf.jpg http://davy.potdevin.free.fr/2%20Roman%20coins%20I%20Augustus%20to%20Vitellius/7-civil%20wars/ric27_1500chf.jpg