The colony of New South Wales was established by an Act of the Parliament (of the Kingdom of Great Britain) in 1787 and in the 24th year of the reign of King George III :
"And whereas his Majesty, by two several Orders-in-Council, bearing date respectively on the sixth day of December, 1786, hath judged fit, by and with the advice of his Privy Council, to declare and appoint the place to which certain offenders, named in two lists to the said several Orders-in-Council annexed, should be transported for the time or term in their several sentences mentioned, to be the eastern coast of New South Wales, or some one or other of the islands adjacent."
Numerous other documents were raised including Phillip's Commission which appointed Arthur Phillip as Captain-General and Governor-in-Chief of New South Wales. :
"To our trusty and well-beloved Arthur Phillip Esquire.
Wee reposing especial trust and confidence in the prudence courage and loyalty of you the said Arthur Phillip of our especial grace certain knowledge and meet motion have thought fit to constitute and appoint and by these presents do constitute and appoint you the said Phillip to be our Captain-General and Governor-in-Chief in and over our territory called New South Wales extending from the Northern Cape or extremity of the coast, called Cape York in the latitude of ten degrees thirty-seven minutes south to the southern extremity of the said territory of New South Wales or South Cape in the latitude of forty-three degrees thirty-nine minutes south and of all the country inland westward as far as the one hundred and thirty-fifth degree of east longitude reckoning from the meridian of Greenwich including all the islands adjacent in the Pacific Ocean within the latitudes aforesaid of ten degrees thirty-seven minutes south and forty-three degrees and thirty-nine minutes south and of all towns garrisons castles forts and all other fortifications or other military works which may be hereafter erected upon the said territory or any of the said islands."
Listen to Ric Hutton reading King George IIIs proclamation at the 1987 Australia Day Ceremony held in Sydney.
Below is a map of New South Wales based on the geographical description detailed above and other charts of the time:
From this map we can see that New South Wales at the time encompassed an area covering the current States of Queensland, New South Wales, Victoria and Tasmania. It also includes large proportions of what is now the State of South Australia and the Northern Territory. By the end of 1825, Van Diemen's Land (now Tasmania) was separated from New South Wales and the western border of New South Wales was relocated to 129 degrees East of Greenwich to include the new settlement of Melville Island.It is interesting to note Section 4 of the current New South Wales State Constitution Act which describes the application of that Act :
"4. Application of Act
For the purposes of this Act, the boundaries of New South Wales shall comprise all that portion of Australia lying between the one hundred and twenty-ninth and one hundred and fifty-fourth degrees of east longitude, reckoning from the meridian of Greenwich and northward of the fortieth degree of south latitude, including all the islands adjacent in the Pacific Ocean within the latitude aforesaid, and also including Lord Howe Island, save and except the territories comprised within the boundaries of South Australia, Victoria, and Queensland, as at present established."
So when we celebrate the 26th of January as Australia Day we do so knowing that the original (colony of) New South Wales encompassed the huge land mass shown in the drawing above. The fact that Western Australia, and parts of South Australia and the Northern Territory were not then part of New South Wales should not impact on the celebration. By comparison we should note that the United States of America (USA) originally consisted of only 13 states and that the other 37 States of the USA readily accept and acknowledge their country's national day despite the fact that they were not part of that original union.
Click below to view a movie of the changing face of Australia's boundaries since 1788:
Governor Phillip, in his first despatch  from the Colony to Lord Sydney wrote:
"We got into Port Jackson early in the afternoon, and had the satisfaction of finding the finest harbour in the world, in which a thousand sail of the line may ride in the most perfect security, and of which a rough survey, made by Captain Hunter and the officers of the Sirius after the ships came round, may give your Lordship some idea.
The different coves were examined with all possible expedition. I fixed on the one that had the best spring of water, and in which the ships can anchor so close to the shore that at a very small expence quays may be made at which the largest ships may unload.
This cove, which I honoured with the name of Sydney , is about a quarter of a mile across at the entrance, and half a mile in length.
We returned to Botany Bay the third day, where I received a very unfavourable account of the ground that was clearing.
The ships immediately prepared to go round, and the 25th - seven days after I arrived in the Supply - I sailed in her for Port Jackson, leaving Captain Hunter to follow with the transports, it then blowing too strong for them to work out of the bay. They joined me the next evening, and all the transports were moored in the cove. "
Russel Ward and John Robertson  describe the events of 26 January 1788 thus:
"In the evening of the 26th (of January) the colours were displayed on shore, and the Governor, with several of his principal officers and others, assembled round the flagstaff, drank the King's health, and success to the settlement, with all that display of form which on such occasions is esteemed propitious, because it enlivens the spirits, and fills the imagination with pleasing presages."
and the proclamation of the colony on 7 February 1788 thus:
"7 February 1788 was the memorable day which established a regular form of Government on the coast of New South Wales. For obvious reasons, all possible solemnity, was given to the proceedings necessary on this occasion. On a space previously cleared, the whole colony was assembled; the military drawn up, and under arms; the convicts stationed apart; and near the person of the Governor, those who were to hold the principal offices under him. The Royal Commission was then read by Mr. D. Collins, the Judge Advocate. By this instrument Arthur Phillip was constituted and appointed Captain General and Governor in Chief in and over the territory, called New South Wales . . . . . "
The Union Flag, or Union Jack, as it is commonly if incorrectly known, is a combination of the flags of England, Scotland and Ireland. It first emerged in a recognisable form in 1603 after the accession of James VI of Scotland to the throne of England. A jack reflecting the unity of the two countries under the king was deemed necessary for shipping, and on 12 April 1606 the first Union Jack, combining the blue-on-white saltire of St Andrew and the red-on-white cross of St George, was used." 
Jack of Queen Anne
This flag is sometimes referred to as the Jack of Queen Anne and is the flag that was raised on 26 January 1788. A plaque commemorating the raising of the flag on 26 January 1788 is located in Loftus Street, Sydney (near Customs House Square, Circular Quay) and was unveiled by the then Governor of New South Wales, His Excellency Air Marshall Sir James Rowland, AC, KBE, DFC, AFC at a ceremony commemorating the founding of the Colony of New South Wales at Customs House Square on Tuesday, 26 January 1988 at 7.30 am. The plaque is located on the pedestal supporting the flagpole from which flies the Jack of Queen Anne. The pedestal, flagpole and flag are located at the closest point to where it is estimated that the Jack of Queen Anne was first raised on 26 January 1788."A saltire which was supposed to represent St Patrick and Ireland was included on 1 January 1801 when Ireland was brought fully into the Union, although this flag was counterchanged to give precedence to that of Scotland."  The Union Jack is shown below and is also seen in the cantons (upper left hand corner) of the Australian National Flag, each of the Australian State flags and other flags flown by the Royal Australian Navy, Royal Australian Air Force and others.
Apart from the Commission, Letters Patent, and other documents there were also George III's Instructions (to Governor Phillip) which covered numerous areas including relationships with the aboriginal people which were quite explicit. 
" You are to endeavour by every possible means to open an intercourse with the natives, and to conciliate their affections, enjoining all our subjects to live in amity and kindness with them. And if any of our subjects shall wantonly destroy them or give them any unnecessary interruption in the exercise of their several occupations it is our will and pleasure that you do cause such offenders to be brought to punishment according to the degrees of the offence. You will endeavour to procure an account of numbers inhabiting the neighbourhood of the intended settlement, and report your opinion to one of our Secretaries of State in what manner our intercourse with these people may be turned to the advantage of this colony."
In 1988 His Royal Highness, The Prince of Wales, accompanied by The Princess of Wales, attended the official Australia Day ceremony in Sydney Australia at which The Prince of Wales gave the official Australia Day speech.
Listen to His Royal Highness, The Prince of Wales give the Australia Day speech on 26 January 1988.
Australia Day 2010
On 26 January 2010, and during his 2010 visit to Australia, Prince William of Wales attended the Governor of Victoria's Australia Day Reception and spoke to the assembled crowd in the State Ballroom. Watch and listen to Prince William's speech below.
(Click on note number to go back to that paragraph)
 NSW Govt Printer (1892), Historical Records of New South Wales, Vol 1, Part 2 (1783-1792) pages 67-70.
 NSW Govt Printer (1892), Historical Records of New South Wales, Vol 1, Part 2 (1783-1792) pages 61-67.
 New South Wales Parliament (1999), Constitution Act 1902 No 32, Section 4.
 Phillip's first despatch from Sydney, Historical Records of New South Wales, Vol 1, Part 2 (1783-1792) page 122.
 After Lord Sydney, Secretary of State, Historical Records of New South Wales, Vol 1, Part 2 (1783-1792) page 121.
 This was the 26th of January, Historical Records of New South Wales, Vol 1, Part 2 (1783-1792) page 122.
 Ward, Russel & Robertson, John (1969), Such was Life - Select documents in Australian social history, Vol 1 (1788-1850), Page 16.
 Shaw, Carol P, (1986), Collins Gem Flags, HarperCollins Publishers, Page 216.