Federation is described in "First Contact" as being 8 000 ly across. Our galaxy is 100 000 ly in diameter, with disk thickness from 2300 to 2600 ly. So volume of Federation should be 1.23 x 10e11 cubic light years. However, from several instances of high-speed transits (DS9-Earth, Earth - Qo'nos) it follows that Federation is not solid block of space, so its volume should be 4.1 x 10e10 to 6.15 x 10e10 cubic light years.
Federation has 150 member planets and 1000 colonies as of early 2370s. Assuming each of member planets had 6 billion population, that gives us 900 billion people living on member worlds... same as expected number of Federation casualties in Dominion War (DS9 "Statistical Probabilities). If each colony had 1 000 000 to 1 billion of residents on average, then total population is 901 billion to almost 2 trillion; if we take "Statistical Probabilities" into accaunt, any number from 1.2 to 2 trillion becomes reasonable, and maybe even more, as we know that some planets have orbital habitats (like Earth).
Also, seeing how population might increase by 2 billion every 20 years, it means that in 2370 human population alone might number 43 billion - possibly even more, due to lots of new planets and Federation science; if we take that into context of colonization, it means that 23rd century colony starting off with 100 000 inhabitants will have 400 000 inhabitants in 24th century, not counting people who come from home planets and larger colonies after establishment of initial colony. This also does not take into accaunt fact that more people you have, population will increase at greater pace; meaning that it is probably low end, especially seeing how new colonies will not be overcrowded as today's Earth is (althought apparent "overcrowding" of planet might simply be due to bad resource management).
CONCLUSION: Federation has 1150 inhabited planets with population count between 900 billion and 2 trillion.
GEOGRAPHICAL SIZE
SW galaxy is described as being "modest" sized galaxy.
-----ANH novel, ch. 6, page 87
"The tridimensional solid screen filled one wall of the vast chamber from floor to ceiling. It showed a million star systems. A tiny portion of the galaxy, but an impressive display nonetheless when exhibited in such a fashion."
Tarkin's speech to Leia from the ANH novel, ch. 7, page 116 :
"This station is the final link in the new-forged Imperial chain which will bind the million systems of the galactic Empire together once and for all."
After Alderaan's destruction, Vader checks the display again. From Chapter 8, page 129:
"Vader stared at the motley array of stars displayed on the conference-room map while Tarkin and Admiral Motti conferred nearby. Interestingly, the first use of the most powerful destructive machine ever constructed had seemingly had no influence at all on that map, which in itself represented only a tiny fraction of this section of one modest-sized galaxy.
It would take a microbreakdown of a portion of this map to reveal a slight reduction in spatial mass, caused by the disappearance of Alderaan. Alderaan, with its many cities, farms, factories, and towns -- and traitors, Vader reminded himself.
Despite his advances and intricate technological methods of annihilation, the actions of mankind remained unnoticeable to an uncaring, unimaginably vast universe." --
From "modest-sized" part we could presume 3200 ly diameter, 32 light years average thickness and 10e9 stars from that - however, some data put minimum diameter for spiral galaxy at 10 000 ly.
We do have something more accurate.
After that, in scene in Jedi Library, we have this display:
From that, we can conclude that Star Wars galaxy has diameter of 12.38 parsecs, or 40.382 light years, and approx. volume of 300 cubic light years.
We know that "million star systems" is tiny portion of galaxy, and that Empire controls "million star systems" (controls, not that these systems are inhabited). So Galactic Empire is not truly "galactic", but rather controls just part of galaxy, which, therefore, could be 10 000 ly in diameter.
NUMBER OF PLANETS
"BAIL ORGANA: It could be a trap.
OBI-WAN: No, I don't think so. The Chancellor will not be able to control the thousands of star systems without keeping the Senate intact."
Which means that Old Republic (and presumably Empire) had "thousands" of inhabited star systems, anywhere from 2 to 19 000. That implies much higher density than Federation has. If we take each world on average having 1 to 5 billion residents (given size of Coruscant, and fact that most Republic and Imperial worlds seemed to be lightly populated, with exception of Coruscant and Alderaan) it could give us 2 to 95 trillion residents. However, given that "hundreds" and then "thousands" of star systems left Republic, anything less than 10 000 inhabited star systems is unlikely, so we have low end for SW population as 5 trillion. That is also supported by "Petition of the 2000"; if we take one senator as representing one system (not unreasonable assumption) that means there are 5000+ inhabited systems. Novelization also gives 1 trillion population count for Coruscant alone.
However, there is also quote from count Dooku in AotC:
"As I explained to you earlier, I am quite convinced that ten thousand more star systems will rally to our cause with your support, gentlemen."
Given that we know that "thousands" of systems already left the Republic, it means that Separatists already number upwards of 2 000 systems; it also means that Republic numbers more than 20 000 systems of upper limit I based on expression "thousands". If 2 000 to 20 000 systems (maybe even 30 to 40 000) already left Republic, and with 10 000 more, we get up to 50 000 initial Separatist systems (More likely 30 000 + 10 000 = 40 000, since any more than 29 000 would be "tens of thousands). It can't go much more than that, however, since these 10 000 systems were obviously counted on as major reinforcement. Republic also probably does not number more than 50 000 to 100 000 systems after Separatists seceded, otherwise disparity would be too great.
While some attempt to portray Empire as having millions of inhabited systems, numbers given in movies (1 200 000 clones etc.) are too low for such numbers. Also, while I avoid T-canon, in TCW series we have mention that Republic will bankrupt if it finances 5 000 000 new clones besides whatever it already has (althought, that probably includes training, hardware, constant supplies - food, water, replacement equipment etc. It is still weird, thought. Given size of population, and assuming army to be only 0.56 % in size - similar to size current proffessional army of Croatia - it should be 11.2 billion soldiers). However, Republic has existed for long time, seemingly without outer enemy, so it could be problematic for it to adapt to war stance.
Also, while there is quote of Empire having 1 million systems in Empire, it is not stated these systems are inhabited, simply that they are there.
POPULATION
We see some planets in canon, which means that we might be able to discern their population.
One is Coruscant - entire planet is covered in skyscrapers. Assuming New York population density we get 553 billion to 5.422 trillion people on Coruscant alone.
Also, you need around 800 acres to feed 1000 people, meaning that there should be 28 to 282 agricurtular worlds with at least 4.42 x 10e11 to 4.3376 x 10e12 acres for Coruscant alone - assuming twice agricultural area per world than that of today's Earth, we have 15.36 billion acres per agricultural world.
In 2008, the world's total arable land amounted to 13,805,153 km², whereas 48,836,976 km² was classified as "agricultural land".
However, other worlds are far smaller - while we don't know much about Alderaan, except that it seems to be similar to modern day Earth (with villages, etc.), which suggests something in order of 4 to 12 billion people, Tatooine and Naboo seem to have population numbering in millions at most; with total population of Republic probably being 5 to 200 trillion.
CONCLUSION: Republic, and Empire after it, have 50 000 - 150 000 inhabited systems and population count of >=5 trillion, up to 200 trillion.
