The Union flag

The Union Flag
When James VI, King of Scots, inherited the thrones of England and Ireland and was crowned James I of England in 1603, the crowns of the Kingdom of England (which included the Kingdom of Ireland and, since the 1535, Wales), and the Kingdom of Scotland were united in a personal union through him. Despite this Union of the Crowns, each kingdom remained an independent state.
On 12 April 1606, a new flag to represent this regal union between England and Scotland was specified in a royal decree, according to which the flag of England (also representing Wales by implication), (a red cross with a white background, known as St George's Cross), and the flag of Scotland (a white saltire with a blue background, known as the Saltire or Saint Andrew's Cross), would be joined together .
Flag of the Kingdom of Great Britain

The Flag of the United Kingdom of Great Britain and Northern Ireland
The current and second Union Flag dates from 1 January 1801 with the Act of Union 1800, which merged the Kingdom of Ireland and the Kingdom of Great Britain to form the United Kingdom of Great Britain and Ireland. The new design added a red saltire, the so-called "cross of Saint Patrick", for Ireland.

The Union Flag is flown from Government buildings at half-mast in the following situations:
from the announcement of the death of the Sovereign (an exception is made for Proclamation Day – the day the new Sovereign is proclaimed, when the Flag is flown at full staff from 11 am to sunset)
the day of the funeral of a member of the British Royal Family
the funeral of a foreign ruler
the funeral of a current or former Prime Minister
The Sovereign sometimes declares other days when the Union Flag is to fly at half-mast. Half-mast means the flag is flown two-thirds of the way up the flagpole with at least the height of the flag between the top of the flag and the top of the flagpole.
The current flag days where the Union Flag should be flown from government buildings throughout the UK are:
20 January (Birthday of the Countess of Wessex)
6 February (Anniversary of the accession of Queen Elizabeth II)
19 February (Birthday of the Duke of York)
Second Sunday in March (Commonwealth Day)
10 March (Birthday of Prince Edward, Earl of Wessex)
21 April (Birthday of Queen Elizabeth II)
9 May (Europe Day)
2 June (Anniversary of the coronation of Queen Elizabeth II)
10 June (Birthday of Prince Philip, Duke of Edinburgh
June (no fixed date) - Official Birthday of Queen Elizabeth II
17 July (Birthday of the the Duchess of Cornwall)
15 August (Birthday of the Princess Royal)
Second Sunday in November (Remembrance Sunday)
14 November (Birthday of the Prince of Wales)
20 November (Anniversary of the wedding of Queen Elizabeth II and Prince Philip, Duke of Edinburgh)

The Union Flag should be flown with the border diagonal band of white uppermost in the hoist (near the pole) and the narrower diagonal band of white uppermost in the fly (further from the pole).
Correct way to fly flag, assuming flagpole to the left
Incorrect way to fly flag, assuming flagpole to the left

The Flag of Scotland
According to legend, in 832 A.D. King Óengus (II) (or King Angus) led the Picts and Scots in battle against the Angles under a king named Athelstan near modern-day Athelstaneford in East Lothian.
King Angus and his men were surrounded and he prayed for deliverance. During the night Saint Andrew, who was martyred on a saltire cross, appeared to Angus and assured him of victory.
On the following morning a white saltire against the background of a blue sky appeared to both sides. The Picts and Scots were heartened by this, but the Angles lost confidence and were defeated.
This saltire design has been the Scottish flag ever since.

Saint Patrick's Cross
The antiquity of the association with Ireland and Saint Patrick has been questioned.
The cross was used in the regalia of the Order of Saint Patrick, established in 1783 as the premier chivalric order of the Kingdom of Ireland, and later in the arms and flags of a number of institutions. After the 1800 Act of Union joined Ireland with the Kingdom of Great Britain, the saltire was added to the British flag to form the Union Flag still used by the United Kingdom.
Saint Patrick's Cross was rejected by Irish nationalists as a British invention.
There is no universally accepted flag for the island of Ireland