SHORT ANSWER
For most of the classical period, non-Greek slaves came primarily from Thrace and Anatolia. There is also evidence of slaves from Syria, Illyria, Macedon and Ethiopia. Slaves of Greek origin were a minority as enslaving fellow Greeks was a sensitive issue.
DETAILS
Non-Greek Slaves
As per the Cambridge World History of Slavery: vol 1,
While statistics are elusive, no one would seriously challenge the
fact that there were large numbers of slaves in the Greek world of
various kinds and origins....The slave trade was everywhere. At the
periphery of Greek culture, slaves were traded all around the Black Sea, in
the Adriatic and in the Eastern Mediterranean and North Africa. So too at
its traditional centres – in Athens, Aegina, Corinth, Chios and elsewhere.
Slaves were bought and sold like any other commodity, and Greek slave traders were active along the coast of the Black Sea, among other areas.
The overwhelming impression created by the ancient literary sources is also that the main supply areas were the regions around the Black Sea (Velkov 1964; Finley 1981, 168–73).
Source: S. Murnaghan, Women and Slaves in Classical Culture
Robin Osborne, in Classical Greece 500 - 323 BC, states that it was Thracians, Anatolians (from Caria, Cappadocia, Phrygia, Lydia etc) and Syrians who were most numerous. In part, this has been deduced from names:
...slaves often bore names which indicated ethnic origins. These names
point to two principal areas of recruitment, the Phrygian, Lydian
and Carian hinterlands behind the Greek cities on the western coast of
Asia Minor on the one hand, and the regions to the north of the
Aegean, principally Thrace, on the other.
Source: Vincent J. Rosivach, 'Enslaving "Barbaroi" and the Athenian Ideology of Slavery'. In 'Historia: Zeitschrift für Alte Geschichte
Bd. 48, H. 2' (2nd Qtr., 1999)
There were also slaves from Scythia, Ilyria (Balkans), Macedon and even (possibly) Iran. For Thrace,
Herodotus (5.6) particularly noted the Thracians’ willingness to sell
even their own children into slavery, and Xenophon (Anabasis 7) refers
several times in his account his adventures in Thrace in 399 to the
activities of slave traders from Byzantium.
Source: S. Murnaghan, Women and Slaves in Classical Culture
Evidence of Ethiopian slaves can be found in the stele shown below from Attica.
"A young Ethiopian groom slave tries to calm a horse down. He holds a whip in his raised right hand and the reins in his left one. Funerary stele from Attica, right side of a two-panels relief. Pentelic marble, found near the Larisis railway station, date unknown (maybe 4th century BC or 1st century BC)." Text & image source. Attrib: [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/ licenses/by-sa/3.0/) or CC BY 2.5 (https://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons
The presence of Celtic slaves in Greece (through trade) is at least plausible as a Greek colony was founded at Massalia (modern day Marseille).
Greek Slaves
Enslavement of fellow Greeks could be a sensitive issue but certainly happened, and it was not just Sparta that did this (and there is some dispute as to whether Helots should be classified as slaves). War was a common source of slaves but
within Greek culture the treatment of the defeated was a major ethical
issue in its own right. While there is scant sign of deep concern
about the fate of defeated ‘barbarians’, the victor had more difficult
decisions to make about defeated Greeks. Enslavement was a real option
and was carried out, but the victor paid a price for that in the
shaping of his reputation, which might well be taken to outweigh any
financial gain, vengeful satisfaction or warning to others
Source: Cambridge World History of Slavery: vol 1
Nonetheless,
When cities fell, there was a recurrent tendency for the victor (even
when dealing with Greeks) to kill the men and enslave the women and
children.
Source: Cambridge World History of Slavery: vol 1
One of the most famous examples is the Athenians enslaving the defeated Melians (women and children - the men were killed) in 416 BC. This led to a lot of negative publicity on Athenian imperialism, hence
the lack of Greek slaves in the historical record for classical Greece
suggests that Greek victors usually preferred another course
Source: Cambridge World History of Slavery: vol 1
Greeks were a minority in the slave population. Some of them were enslaved through war (though at least some enslaved through war were sold abroad), while others fell into slavery because of poverty and debts. Worth mentioning are Spartan nurses:
presumably helot women, had a high reputation outside
Sparta, so high that Alcibiades of Athens, for example, was reared by
such a Spartan helot nurse.
At the end of Peloponnesian war, some of Sparta's allies wanted the defeated Athenians to be enslaved but Sparta resisted this, arguing that Athens' contribution to the defeat of Persia some 100 years earlier should not be forgotten. Also, the Spartans probably felt that Athens was needed to counter the growing influence of Thebes.
A FEW OTHER POINTS OF INTEREST
It is also worth noting that both Greek and foreign slaves could be freed by being ransomed - an often lucrative source of income for slave traders.
When it comes to numbers,
the Greeks had little idea how many slaves there were and estimates of
their numbers seem greatly exaggerated. The census of Demetrius of
Phalerum, for instance, held between 317 and 307, counted 400,000
slaves in Athens. Modern scholars consider even a quarter of this a
high estimate.
Around the late fifth century, there were an estimated 80,000 to
100,000 slaves in Attica of which between 10,000 and 20,000
worked in the silver mines at Laurium. The Athenian general Nicias
owned 1,000 slaves working in the mines, according to Xenophon.
Source: R. Osborne
Other sources
Paul Cartledge, Sparta and Lakonia: A Regional History
https://cliojournal.wikispaces.com/Slavery+in+Ancient+Greece
M. Gann & J. Willen, Five Thousand Years of Slavery
