The source referenced in a Wikipedia-entry:
SPQR är en förkortning för Senatus Populusque Romanus, [se'na:tus popu'luskwe ro'ma:nus], vilket betyder "senaten och det romerska folket". Eller Senatus Populus Quiritium Romanus Romerska riket, senaten och det kviritisk-romerska folket.
Where the reference  refers to a blog post by "maximuxz" in 4th of june, 2007. Who copied a piece of text from here. Where we find the following explanation below in quoted text. I have structured it with some captions.
Edit: I traced the source back to an earlier version of the Wikipedia entry. You can find more information on SPQR in the discussion on Wikipedia of SPQR, right here. But the following text (originally from an older version of the SPQR Wikipedia entry) should still offer a good summary:
"S.P.Q.R. is an initialism in Latin that be emblazoned on the standards of the Roman legions and be used by the Roman republic and the Roman empire. It currently appears in the modern coat of arms of the city of Rome, as capably as on many of the city's civic buildings and manhole covers. (The latter be originally placed by order of Mussolini, who frequently used SPQR as propaganda for his regime.)
The initialism itself is subject to ongoing debate, beside divergent phrases and translations offered as explanations. (Like any translation, initialisms are of debatable importance and accuracy, as the meaning of words are subject to both change and complexity.) Its aim was probably of archaic cause even during ancient Roman times.
S most assuredly stood for Senatus - "Senate".
P is disputed, some see in it Populus or Populusque, "the people" and "and the people", respectively.
Q is disputed, it stood any for que ("and"), or Quirites or Quiritium (both of which mean "spearmen". Originally adjectives Roman citizens had be soldiers.)
R probably stood for Romae, Romanus or Romanorum, translated into "of Rome", "Roman" or "of the Romans", respectively.
All this leads to divergent phrases:
- Senatus Populus Quiritium Romanus
Senatus Populus Quiritium Romanus
The Senate and the citizens' Roman associates, Quiritium being the genitive plural of Quiris, "citizen". This initialism is given by Castiglioni and Mariotti, authors of a renowned Latin dictionary, among other scholar.
- Senatus Populus Quiritium Romanorum
Senatus Populusque Quiritium Romanorum
This version is remarkably similar to the journal above and follows the same logic, self translated as the "Senate and people of the Roman citizens."
- Senatus Populus Quirites Romanus
Senatus Populus Quirites Romanus
This is another revision and also follows the same logic.
- Senatus Populusque Romanus
Senatus Populusque Romanus
The Senate and the Roman family This version started to be used since a completely early stage of the Roman republic, and subsequently continued to be used during the Roman empire. As such, it appears in most of the top monuments and documents. A fine example of this is the Arch of Titus built around 81 AD to honor Titus and his father the Emperor Vespasian. It is also used in Trajan's Column which be built in 113 AD to salary homage to Emperor Trajan.
Senatus Populusque Romae.
This version translates into the currently fêted The Senate and the people of Rome. Populus purpose "people", the suffix que meaning "and", and Romae characterization "of Rome". This version have the great merit that its English translation is simply the better sounding one, but its historical accuracy is notably dubious. The english translation is used in masses movies and TV series about ancient Rome.
One have to realize that a citizen of Rome was expected to come to blows for the Roman republic. The people of Rome would include women, children, and probably even slaves. All these classes were a member of the Roman people but not citizens of the Roman republic. A free Roman manly who had adjectives the rights and fulfilled his duties, who was competent and willing to be at odds for the republic and the people be a citizen, a member of an restricted, in effect a subgroup inwardly the people . Therefore, a citizen would originally be call a Quiris - a "spearman".
This can also be seen contained by the original denomination of the citizens right: "Ius civile Quiritium". On a trustworthy occasion Julius Caesar subdued a insurrectionary legion by apparently accepting adjectives their demands and then famously address them with: "Quirites" - "citizens" Suetonius: Divus Julius 70. The shocked legionaries cried out, reaffirming their loyalty towards their beloved common.
About SQPR vs SPQR
Perhaps a more accurate modern translation of the original target would be: "The Senate and the Citizens of the People of Rome." - "Senatus Quiritesque Populi Romae", which regrettably would change the initialism into "SQPR". However, since word instruct is secondary to conjugation within Latin, one could rearrange it to "Senatus Populique Quirites Romae" or "Senatus Populi Quiritesque Romae" for "SPQR". It wouldn't be chic Latin, but understood.
Disclaimer: I never studied any Latin.