The referendum question was whether to accept an amendment to Article 40 of the Irish Constitution so that provision can be made in law for the regulation of termination of pregnancy. The Irish electorate voted by 1,429,981 votes to 723,632 in favour of amendment - i.e. 66.4% to 33.6% with a turnout of 64.13%.
Ireland has a written (formal) constitution dating from December 1937. The Constitution includes fundamental rights (Articles 40-44) and, in particular, Article 40.3.3 -
"The State acknowledges the right to life of the unborn and, with due regard to the equal right to life of the mother, guarantees in its laws to respect, and, as far as practicable, by its laws to defend and vindicate that right."
Changes to the Constitution itself have to be made by a process which includes a referendum and Article 40.3.3. came about following a referendum held in 1983. The effect of Article 40.3.3 - as interpreted by the courts - has been to prevent lawful abortion in Ireland in most circumstances. The present legal position is that it is lawful for a pregnancy to be terminated only where it poses a real and substantial risk to the life of the mother, including a risk of suicide.
In March 1992 the Irish Supreme Court decided Attorney-General v X and others. A Timeline of the X case and the Judgment is available via the link HERE. In X, a 14 year old girl became pregnant as a result of being raped by a man known to her and her family. She had been sexually abused by the same man for some 2 years. X and her parents decided to travel to the UK so that X could have an abortion. The family asked the Irish Police (Gardai) whether, after the abortion, the foetus could be tested to provide proof of paternity. The Police asked the Director of Public Prosecutions whether such evidence would be admissible in court. In turn, the DPP liaised with the Attorney General who obtained an injunction stopping X and her parents leaving the country or arranging for termination of the pregnancy. Ultimately, the Supreme Court discharged the injunction thereby permitting the threat of suicide as a ground for abortion.
In December 1992, the Constitution was amended so that the right to life of the unborn did not affect freedom to travel between Ireland and another State (Amendment 13) and also did not limit freedom to obtain or make information available relating to services lawfully available in another State (Amendment 14). UK government statistics for 2016 show that, for women non-resident in the UK, there were 4810 abortions and 68% of those were from the Republic of Ireland and 15% from Northern Ireland.
In 2013, Ireland gave legislative effect to the X case - see the "explainer" at thejournal.ie. The legislation is the Protection of Life During Pregnancy Act 2013.
The outcome of the 2018 Referendum is that Article 40.3.3 will be replaced by a much simpler Article stating - "Provision may be made by law for the regulation of termination of pregnancy." Legislation is expected before the end of the year to amend the law.
Contrast with the UK:
By contrast with Ireland, the United Kingdom does not have a written (or formal) constitution. Rights are not guaranteed in the UK and are at the mercy of Parliament. The Human Rights Act 1998 is the closest one comes to human rights protection since the Act enables the courts to make a declaration that legislation is incompatible with "convention rights." The making of a declaration does not alter the law but signals to Ministers / Parliament the need for change. Nevertheless, Parliament is not obliged to amend the law though the Human Rights Act contains a "fast-track" procedure for doing so..
In Ireland, constitutional amendments have been required to permit ratification of various European Union Treaties - e.g. the 28th Amendment dealt with the Lisbon Treaty. On each occasion, a referendum was held to approve the amendment.
For the UK, the European Union Act 2011 made provision for referendums in connection with any treaties replacing the Treaty on European Union or the Treaty on the Functioning of the EU. This Act will be repealed when the European Union (Withdrawal) Bill becomes law.
In the UK Supreme Court, the Northern Ireland Human Rights Commission challenged the Northern Ireland legal position claiming that the law is not compatible with the European Convention on Human Rights because it fails to provide an exception to the prohibition on the termination of pregnancy in Northern Ireland in cases of serious malformation of the unborn child/foetus or pregnancy as a result of rape or incest. See Supreme Court - NIHRC Application for Judicial Review The court heard argument over 24th-26th October 2017 and judgment is awaited. The case highlighted situations such as that of Ashleigh Topley who became pregnant with a much wanted child. A scan at 20 weeks revealed that the baby had a fatal abnormality but the pregnancy had to run to full term because of the law in Northern Ireland - The Independent 4th July 2017.
Justice Gap 30th October 2017 - Abortion law in the UK is plagued by hypocrisy and cowardice. Nowhere more so than in Northern Ireland.
In 2017, the UK Supreme Court - by a 3 to 2 majority - held that it was lawful for the NHS in England not to provide funding for abortions for those from Northern Ireland - R (on the application of A and B) v Secretary of State for Health  UKSC 41. The case is discussed in some detail on the Bristol University Law School blog. Funding for women from Northern Ireland became available in England as a result of action in Parliament. This was followed in Wales and in Scotland.
The consent of the Irish people is needed before the Constitution can be amended. This means that amendments to the Constitution can only be made by way of a constitutional referendum. To date, 35 constitutional referendums have been held, of which 29 resulted in amendments to the Constitution. A list of the amending Acts is at the front of the Constitution published on irishstatutebook.ie.
Read more about constitutional referendums.