A letter rogatory or letter of request is a formal request from a court to a foreign court for some type of judicial assistance. The most common remedies sought by letters rogatory are service of process and taking of evidence.
Service of process 
Courts may serve documents only to individuals within the court's jurisdiction. One exception to this rule is states that invoke universal jurisdiction, granting their courts ubiquitous domain. Therefore a person seeking to take an action against a person in another country will need to seek assistance from the judicial authorities in the other country. This is of course assuming the court in his own country has jurisdiction to hear the case matter. As a hypothetical example, Alice in the United States wishes to sue Roberto in Argentina. Alice issues her summons in a U.S. court, and must then petition a court in Argentina by means of a letter rogatory to serve the process on Roberto.
The use of letters rogatory for purposes of service of process to initiate court action is now largely confined to the Americas, as between countries in Europe, Asia, and North America, service of process is effected without resort to letters rogatory, under the provisions of the Hague Service Convention.
Taking of evidence 
Another reason why a court may require assistance from a foreign court is to obtain evidence from a witness. This evidence may be to answer questions relevant to the determination of an issue of fact, or for disclosure of documents.
Courts only have power to subpoena witnesses from within their own country. So for example Alice in the U.S. could not summon Jean from France to the U.S. courthouse. Instead the U.S. court would issue a letter rogatory to a French court, who would then examine Jean in France, and send a deposition back to the requesting court.
Insofar as requests to United States courts are concerned, the use of letters rogatory for requesting the taking of evidence has been replaced in large part by applications under 28 U.S.C. section 1782, or Section 1782 Discovery.
In many cases, the witness is willing to provide the testimony. However, the target court may compel the testimony of a witness who is unwilling to appear.
Generally, the target court will agree to the request unless it violates a policy of the target jurisdiction. For example, in the United States, it is usually appropriate in a civil case to depose every available witness, while in Canada, you may only depose one witness from each party. However, a Canadian court would most likely agree to the pre-trial deposition of a witness who could be compelled to testify in the United States.
In the past, letters rogatory could not usually be transmitted directly between the applicable courts, and had to be transmitted via consular or diplomatic channels, which could make the whole process very slow. There have been various international conventions in regard to service of process and taking of evidence. One of the earliest conventions to simplify the procedure of Letters Rogatory was the 1905 Civil Procedure Convention, signed at The Hague. Drafted only in French, it was only ratified by 22 countries. Later conventions, created after the institution of the Hague Conference on Private International Law, drafted in both English and French commanded more support. The Hague Service Convention, ratified in 1965, enabled designated authorities in each of the signatory states to transmit documents for service to each other, bypassing the diplomatic route. This convention has been ratified by 60 states including the United Kingdom and the United States, neither of whom had ratified the 1905 convention. The Hague Evidence Convention, ratified in 1970, formalised procedures for taking of evidence. This convention has been ratified by 43 states. For situations exclusively among Member States of the European Union, two regulations, 1348/2000 and 1206/2001 superseded the two Hague Conventions. These two regulations apply to each of the Member States of the European Union with the exception of Denmark which opted out.
For countries not signed up to any convention, the letter rogatory is still used. So for example, a plaintiff in the Netherlands which is signed up to the Hague Service Convention would still need to issue a letter rogatory to Brazil, as the latter has not signed the Convention.
The European Convention on Mutual Assistance in Criminal Matters (CETS No. 030), open for signature by the Members of the Council of Europe, in Strasbourg, on 20 April 1959, stipulates that parties agree to afford each other the widest measure of mutual assistance with a view to gathering evidence, hearing witnesses, experts and prosecuted persons, etc.
The Convention sets out rules for the enforcement of letters rogatory by the authorities of a party ("requested party") which aim to procure evidence (audition of witnesses, experts and prosecuted persons, service of writs and records of judicial verdicts) or to communicate the evidence (records or documents) in criminal proceedings undertaken by the judicial authorities of another party ("requesting party").
The Convention also specifies the requirements that requests for mutual assistance and letters rogatory have to meet (transmitting authorities, languages, refusal of mutual assistance).