Temporarily places a diocese directly under the pope. This allows the pope to act directly through his administrator, rather than through a bishop. An Apostolic Administrator can move in to take over a Church province, which can offer a quiet way of firing a bishop.
(Latin: administratura apostolica)
This temporarily removes a Church territory from a local bishop and places it directly under the pope. This has two functions: (1) to reorganise Church territories so that they correspond with national ones or (2) to fire a bishop.
1. As an interim measure before synchronising Church and national boundaries
As the Vatican's Secretary of State Luigi Maglione said, shortly after the outbreak of World War II, "the Holy See cannot run behind armies and change bishops as combat troops occupy new territory belonging to countries other than their own".  An interim solution was to appoint an apostolic administration.
This was done in the aftermath of World War I when national boundaries were redrawn. For instance, an apostolic administration is mentioned in (Austrian concordat 1933, Art.3.2) There it was used as an interim measure for Innsbruck which had originally belonged to the Diocese of Brixen before most of that was ceded to Italy.
The Vatican also used apostolic administration on a widespread basis during World War II. Controversy was cause by the way Pius XII set up an apostolic administration for the Polish "Recovered Lands", former German territory which had been granted to Poland at the end of World War II by the international peace conference at Yalta. The pro-German Pius XII was reluctant to recognise this, as shown by his letter to the German bishops on 1 March 1949.  After that "it took Pope Paul VI until 1967 to appoint Polish apostolic administrators, and until 1972 to establish formal Polish dioceses when the two countries ratified a border treaty."  In other words, here, too the appointment of an apostolic administration marked the (belated) beginning of the process of reconciling Church boundaries with national ones.
2. For firing a bishop
♦ An apostolic administration was used in 1955 to remove the tactless Spanish Cardinal Segura who lambasted "freedom of religion" and lauded the "meritorious Inquisition", etc. 
♦ It also came in handy for taking the administration out of the hands of the Australian Bishop William Morris. In 2011 he was relieved of his duties after suggesting in a pastoral letter the ordaining women and married men to the priesthood to solve the shortage of priests in the Diocese.of Toowoomba. 
♦ Two years later, on the last day of his papacy, Benedict XVI appointed an apostolic administration to fill in for Cardinal Keith O'Brien after the Scottish prelate publicly recommended that the Church let priests marry. 
♦ And in 2014 Francis I appointed an apostolic administrator to take over from "the bishop of bling". Franz-Peter Tebartz-van Elst, Bishop of Limburg in Germany, had become a liability. He had the Church to pay a fine on his behalf, rather than face perjury charges.  And he also ran up a bill of €31 million for his bishop's residence.  With the apostolic administrator in place, the ex-bishop was packed off to a monastery for a "spiritual time of recovery" of unspecified length. 
1. Pierre Blet, and Lawrence J.Johnson, Pius XII and the Second World War: According to the Archives of the Vatican, 1999, p. 73.
2. "On March 1, 1949, pope Pius XII sent a letter to the German Episcopate announcing that the transfer of German territory in the East to Poland was still an open issue. This again worsened the relations between state and church. The decree of the Holy Office of July 13, 1949, that anathemised Roman Catholics who collaborated with the Communist authorities did not help the situation."
Pawel Zalecki, "The Roman Catholic Church", Europe Since 1945: An Encyclopedia, ed. Bernard A. Cook, 2001, Google reprint.
3. Jonathan Luxmoore, "Europe's reparation 'war'", The Tablet, 2009-02-28. http://www.thetablet.co.uk/article/1815
"In 1945, with tacit Vatican consent, the Polish [Catholic] Church took over the running of German parishes in the 'Recovered Territories'. It took Pope Paul VI until 1967 to appoint Polish apostolic administrators, and until 1972 to establish formal Polish dioceses when the two countries ratified a border treaty."
4. "Religion: Unemployed Archbishop", Time Magazine, 1955-11-21. http://www.time.com/time/magazine/article/0,9171,866644,00.html
5. "Australian Bishop Removed; Supports Women Priests: Apostolic Administrator Named for Toowoomba", Zenit, 2011-05-02. http://www.zenit.org/article-32456?l=english
6. "Apostolic Administrator Chosen for St. Andrews and Edinburgh", Zenit, 2013-02-28. http://www.zenit.org/en/articles/apostolic-administrator-chosen-for-st-andrews-and-edinburgh
Catherine Deveney, "Cardinal Keith O'Brien: how Britain's Catholic leader fell from grace", Observer, 2013-03-02. http://www.guardian.co.uk/world/2013/mar/02/cardinal-keith-obrien-sex-scandal-priests
7. "Case Closed: Court Stops Proceedings Against 'Bishop Bling'", Spiegel, 2013-11-18.
8. "Vatican Removes Germany's 'Bishop of Bling'", Reuters, 2014-03-26. http://www.nytimes.com/reuters/2014/03/26/world/europe/26reuters-vatican-germany.html
9. "Under-fire German bishop retreats to monastery", Seattle Times, 2013-10-31. http://seattletimes.com/html/nationworld/2022162389_apxgermanybishopsspending.html