As other answers note, τῷ πραιτωρίῳ in Phil 1:13 may refer to either a place (the governor's palace) or people ("praetorian guard"), or in a more extended sense, the wider household (all this from Liddell-Scott).
What is not noted is where this "palace" or "guard" or "household" was located geographically (which I take it is the main point of OP's question). The nuance ascribed to πραιτώριον here has some bearing on this matter, but since most translations and commentators take this as a "person" reference (typically, "praetorian guard"), we'll move ahead on that basis.
The location from which Paul wrote and sent this letter to the church at Philippi is still debated. The evidence available is restricted to: (a) the internal evidence of Philippians itself; (b) whatever light Acts sheds on Paul's movements and career (itself a disputed source, but still used carefully by mainstream biblical scholars); and (c) wider evidence of the Roman world. There are four options canvassed:
This has had some recent defenders, but is the least plausible of the four. It is purely conjectural, attempting to meet the requirements of internal evidence ("(a)", above; but was he actually imprisoned there?), and is noted here just for the sake of completeness.
The Ephesian option has gained some popularity in recent years (especially, so far as I can tell, among some German scholars), and is taken seriously by the weightier commentators. Like Corinth, though, it is purely conjectural; it has the further difficulty of not clearly satisfying Phil 1:13, as there seems no solid evidence that there actually was a "praetorian" presence in Ephesus.
This option has been championed in recent years by Gerald Hawthorne, both in his Word Biblical Commentary, and in the entry he provided for the Dictionary of Paul and His Letters. Paul in Caesarea has a lengthy custodial sentence, the presence of a praetorium, and access to friends -- although Timothy's presence remains inferred.
Traditionally, based in part on this very verse (Phil 1:13), Rome has been assumed to be the place of writing and sending. OP's verse is supplemented by the closing greeting:
All the saints greet you, especially those of Caesar's household. (Phil 4:22, ESV)
which has naturally inclined thoughts towards Rome, although this is not strictly required by these words.
The major strike against Rome (that has occasioned the search for other possibilities) has been geographical distance. Internal evidence requires some to-and-fro of communication between Paul and the Philippian Christians, and Rome was thought to be simply too far away for this to be plausible. More recent studies (and that of Stephen Robert Llewelyn, "Sending Letters in the Ancient World: Paul and the Philippians", Tyndale Bulletin 46 (1995), 337-356, has been influential) suggest that this is not the obstacle some have made it, removing it as a strike against Roman provenance.
As noted, this remains a debated question, but it seems the traditional association with Rome persists as being the one mostly favoured by commentators, e.g.:
- Gordon D. Fee, Paul's Letter to the Philippians (NICNT; Eerdmans, 1985), pp. 34-37;
- Moisés Silva, Philippians, (Baker Academic, 2005), pp. 5-7;
- Ben Witherington, Paul's Letter to the Philippians: A Socio-Rhetorical Commentary (Eerdmans, 2011), pp. 9-11.
- + older commentators, H.A.A. Kennedy in the Expositors Greek Testament, vol. 3 (1895); and M.R. Vincent in the International Critical Commentary (1897).
- F.F. Bruce, "St. Paul in Macedonia, 3, The Philippian Correspondence", Bulletin of the John Rylands Library 63 (1981): 260-284;
- Bo Reicke, “Caesarea, Rome, and the Captivity Epistles”, in W. Ward Gasque & Ralph P. Martin, eds., Apostolic History and the Gospel: Biblical and Historical Essays Presented to F.F. Bruce (Exeter: The Paternoster Press, 1970), pp. 277-286.