Doctor of Divinity
Doctor of Divinity (D.D. or DD, Divinitatis Doctor in Latin) is an advanced or honorary academic degree in divinity.
Contrast with other religious degrees
Doctor of Divinity is not to be confused with Doctor of Theology (Th. D.) which is a research doctorate, in theology, awarded by universities and divinity schools such as Harvard Divinity School and many others. Many universities award a Ph.D. rather than a Th.D. to graduates of higher-level religious studies programs. Doctor of Sacred Theology is a research doctorate in theology, but particular to Catholic Pontifical Universities and Faculties. Doctor of Ministry is another doctorate-level religious degree, but is a professional rather than a research doctorate. 
Doctor of Divinity by country or church
In the United Kingdom, Doctor of Divinity has traditionally been the highest religious doctorate granted by universities, usually conferred upon a religious scholar of standing and distinction. It is a higher doctorate generally awarded for accomplishments beyond the Ph.D level. 
The Doctor of Divinity degree is awarded in recognition of a substantial body of original research undertaken over the course of many years. Typically the candidate will submit a collection of work which has been previously published in a peer-refereed context and pay an examination fee.  The university then assembles a committee of academics both internal and external who review the work submitted and decide on whether the candidate deserves the doctorate based on the submission. Most universities restrict candidacy to graduates or academic staff of several years' standing.
In the United States, Doctor of Divinity is traditionally an honorary degree granted by a church-related college, seminary, or university to recognize the recipient's ministry-orientated accomplishments. 
Most American universities award Doctor of Divinity only as honorary degrees. American universities do not have the system of higher doctorates used in the UK and some other universities around the world. Martin Luther King was a Doctor of Divinity.
Because there are no requirements, any organization may grant a Doctor of Divinity degree, and some will do so to anyone for a small fee. This right was affirmed in 1974 in Universal Life Church, Inc. vs. United States, 372 F. Supp. 770, 776 (E.D. Cal 1974) 
|“||The fact that the plaintiff distributed ministers credentials and Honorary Doctor of Divinity certificates is of no moment... Neither this Court, nor any branch of this Government, will consider the merits or fallacies of a religion. Nor will the Court compare the beliefs, dogmas, and practices of a newly organized religion with those of an older, more established religion... Were the Court to do so, it would impinge upon the guarantees of the First Amendment.||”|
—Judge James F. Battin, Universal Life Church, Inc. vs. United States, 372 F. Supp. 770, 776 (E.D. Cal 1974)
The Doctor and Student
Christopher St. Germain's 1528 book The Doctor and Student describes a dialogue between a Doctor of Divinity and a law student in England containing the grounds of those laws together with questions and cases concerning the equity thereof.