According to Debrett's Baronetage, "The term Baronet was first applied to the nobility who lost the right of individual summons to Parliament; and in this sense was used in a statute of Richard II. It is said that Sir Robert Cotton's discovery of William de la Pole's patent in the 13th year of Edward III, conferring upon him the dignity of a Baronet in return for a sum of money, suggested the revival of the Order."

The hereditary Order of Baronets in England was erected by Letters Patent (i.e. At the express wish of the sovereign, rather than as the determination of the Lords or Parliament) by King James I on 22 May 1611, "for the settlement of Ireland." or, according to Encyclopaedia Britannica,"in order to raise some money. Because the money was ostensibly for support of the troops in Ulster, candidates for the baronetage were required to pay the king £1,095 (the sum required to maintain 30 soldiers for three years)"
According to Debretts, he offered the dignity to 200 gentlemen of good birth, with a clear estate of £1,000 a year, on the condition that each one should pay into the Kings Exchequer in three equal installments a sum equivalent to three years' pay to 30 soldiers at 8d per day per man. The first installment was to be paid on the delivery of the patent.

Burke's explains that is why the badge of the Red Hand of Ulster features as a baronets device (except in the case of Nova Scotia creations, where the saltire of Ulster was used instead.) At the same time the Red Hand is not invariably shown in a baronet's coat of arms.
At the same time, it was made clear that no order should henceforth come into existence that was of equal or higher degree than the baronetage yet beneath the lowest rank of peerage. [Encyclopaedia Britannica points out that a Baronet in England ranks below a Knight of the Garter, and in Scotland, below a Knight of the Garter and a Knight of the Thistle.]

Another batch of Baronets, the Baronetage of Ireland was erected on the 30th September 1611 and on the 28th of May, 1625 the Baronetage of Scotland or Nova Scotia was erected, with each Baronet required to pay £2000 (the amount required ostensibly for the support of six Nova Scotian colonists), and also to pay a fee of £1,000 to Sir William Alexander, who had been granted the province in 1621.
The Scottish Baronetage was, like the earlier two, set up by James I, but carried on by Charles I. The Scottish Baronets got a few extra goodies for their money - the right to wear a neck badge, and, before 1638[or 1683,in Debretts], a grant of 16,000 acres of the plantation of Nova Scotia, with plenary baronial rights and jurisdiction, and legislative powers in that plantation. They were also given precedency above the lesser Barons in Scotland, the right to add the Arms of Nova Scotia to their armorial bearings (instead of the Red Hand of Ulster), and the power to sit and vote by deputy in the Scottish Parliament when absent from the Kingdom.

The English Baronets had (in their first Letters of Patent) the comfort of knowing that only two hundred Baronets of England were to exist at one time - but that exclusivity was soon withdrawn and replaced with the requirement that no degree or dignity was ever to be created which would be superior or equal to the degree and dignity of a Baronet.
Under the second Letters Patent it was stated that no person or persons should have place between Baronets and the younger sons of Viscounts and Barons, and that Baronets and their eldest son or heir apparent (on coming of age, at 21 years old) had the right of knighthood (this compensation was included because candidates for Baronetcies had dried up following his ruling that baronets' precedence should be lower than that of barons' younger sons. Also, that would mean William Elliot is styled esquire rather Sir through his own choice), and they get to include the badge of the Red Hand of Ulster in their armorial bearings. A third Letters Patent ratified all the privileges of the second, and in addition gave the eldest son of a Baronet precedence over the eldest son of Knights of whatever order.
Irish Baronets had similar privileges to English ones, only they used a dexter hand for the Badge of Ulster. All baronets were accorded the courtesy title of Honourable from the erection of the degree, but this fell into disuse at the beginning of the 19th century. A Baronets wife could be called 'Dame Christian name' (now used only in legal and formal documents) or 'Lady Surname', same as a knights lady. The helmet of a Baronet is like a knights, too, depicted full faced, with the visor up and without bars.

As you could probably guess, the English and Irish Baronets coveted the additional privileges of the Nova Scotia and Scottish Baronets - from the time of Charles I, Baronets of England and Ireland were petitioning the King for the right to wear a badge around their necks. Some of them wore badges around their necks anyway in the 17th century, and in 1834 Sir Richard Broun, later 8th Bt, was still petitioning for the honour - Disraeli immortalized him in Sybil as Sir Vavasour Firebrace for his pains. Finally, in 1929, George V accorded all Baronets their own neck badge)

The destination of a baronetcy is in accordance with the limitation mentioned in the patent at creation. Though this was usually to the heirs male of the body of the first baronet, special remainders were sometimes granted (which would appear to be the case with the Elliots, if 'of the body' = 'son').
Baronetcies of Nova Scotia, or Scotland, were sometimes created with remainder to heirs male (that is, the heir male to the first baronet, despite the extinction of his own descendants in the male line) and 'heirs male and of tailzie' (only four exist today, this means the descent together with entail of estates, may pass to the heir general.)
Children adopted into a family do not gain rights of succession to a title, and children adopted out of a family do not lose their rights.

Knights, like Sir William, had to be presented at Saint James to be dubbed on special occasions (such as Royal Weddings), and was preceded by various rituals like the presentation of robes, arms, spurs, vigils, baths, depending on the order of the knighthood.

Strangely, the details I could find of money for honours all stopped at Charles I. There was some mention of needing a certain amount of land to remain qualified for a Baronetcy before 1909, in the 1935 Bourkes Landed Gentry, but I am not sure if this was a specific requirement for Baronets, or part of a convoluted whinge about land taxes in the first part of the twentieth century.
According to Burke's Baronetage, in the 19th century, the custom grew up of conferring baronetcies on distinguished men who were deemed not quite worthy of a peerage (due to their calling, rather than their wealth) - Engineers, medical men, Lord Mayors of London. Still, in courts as infamously corrupt as that of George III, I can't help suspecting money helped a candidate's eligibility, especially as he holds the record for creating Baronets (469, or about 30% of all 1529 creations from 1616 to 1818). Also, I think there were restrictions on the Prince Regent's abilities to create Baronets, until he ruled in his own right (although he seemed to be able to accord a baronetcy to the Lord Mayor of London in 1814), and even then he only created 55.
On how they were selected, Debrett's 7th edition (from 1835) notes that candidates were at first "carefully selected from the most wealthy and distinguished families of landed gentry. In subsequent periods although landed property and interest has always been much regarded in these promotions, many other considerations arising out of the circumstances of the times and the progress of society have been allowed to direct the royal favour. In the reign of the first Charles it was at once the stimulus and reward of devoted loyalty, by his son it was often bestowed as an honorary recompense for sufferings and attachment on which he was unable or unwilling to confer more solid remuneration. As the commerce of our country increased the wealthy and distinguished Merchant learned to look forward to it as the well earned meed of long continued and successful industry." There is also mention of the honour going to some of the "heros of the Nile and Trafalgar and Waterloo" ( Mary Musgrove must have been relieved to learn that it would be Admiral Croft, not Captain Wentworth, in line for a Baronetcy, although I am not sure Sir Walter would like this!).
After the Act of Union in 1707, Baronets were created for Great Britain, rather than for Scotland or England, and after 1801, Scottish, English and Irish creations were all Baronets of the United Kingdom.

The first Baronet created after 1815 was Sir Walter Scott, gazetted as the 30th of March 1820. Upon receiving the first proposal of the honour in 1818, he confided in Johanna Baillie "I have now before me Lord Sidmouth's letter containing the Prince's gracious and unsolicited intention to give me a Baronetcy. It will neither make me better nor worse than I feel myself in fact it will be an incumbrance rather than otherwise but it may be of consequence to Walter [his son, who predeceased him] for the title is worth something in the army, although not in a learned profession. The Duke of Buccleuch and Scott of Harden who, as the heads of my clan and the sources of my gentry are good Judges of what I ought to do, have both given me their earnest opinion to accept of an honour directly derived from the source of honour and neither begged nor bought as is the usual fashion. Several of my ancestors bore the title in the 17th century and were it of consequence I have no reason to be ashamed of the decent and respectable persons who connect me with that period when they carried into the field like Madoc:
'The crescent at whose gleam the Cambrian
oft Cursing his perilous tenure wound his horn'"
In an even earlier letter to Mr Richardson, 22nd August 1819, Scott reveals that he has hired a herald called Hardon to draw up a coat of arms in preparation for the event, which he sends to Richardson with a copy of his seal, and advises him to "furnish Sir George Naylor with as much of my geneology as will serve the present purpose", he adds that he should not mind if some small expenses are made for this purpose, and also sends a geneology he has made of his fathers and fathers grandmothers decent with "memorandums of proofs by which they may be supported", and he claims that in Edinburgh he is making it his business to be of service to his granduncle, "the last Haliburton of Dryburgh Abbey", and consulting with "the Lyon-Office people in Edinburgh",so it seems there was a few professionals to pay for the special bits and pieces needed for a Baronetcy, and representatives to deal with well before time. Scott's son-in-law and heir, John Gibson Lockhart (who did not inherit the Baronetcy) claims "Scott's baronetcy was conferred on him not consequence of any ministerial suggestion but by the King personally and of his own unsolicited motion and when the Poet kissed his hand he said to him 'I shall always reflect with pleasure Sir Walter Scott's having been the first creation of my reign'"

However, other sources remember it differently and while neither Lockhart or Scott are entirely trustworthy historians, they were both influential men of the red-hot Tory press, and well connected lawyers, skilled at giving and taking sinecures and pensions, shaping interest and trading in the Regency equivalent of CDO's (bills of trade), as people of their professions did in those days. I can see for myself some differences between their representation of the process and what the Baronetages claim (for example, Lockhart's boast that the honour was granted by the King himself, and not on the advice of a minister, when the Baronetage makes it clear that this has always been the case for all Baronets). On the other hand, some of the 'history' that found its way into the Baronetages was less credible than Lockhart's and Scott's.

Debrett's first Baronetage was published in 1808, but there were a number advertisements in the newspapers of the time for now extinct Baronetages (or rather, in abeyance because Baronetcys fall into abeyance rather than become extinct, as Peerages do. Also unlike a Peerage, there's no method of formally renouncing a Baronetcy, and if a young heir-presumptive should choose not to take the honour he inherits, or succession is not obvious, the Baronetcy is said to be dormant.) The Official Roll of the Baronetage kept at the Home Office, which establishes who is entitled to be received as a Baronet and mentioned by that title in any civil or military commission, letters patent or any other official document, did not come into being until the 8th of February 1910.

References: Debrett's Peerage and Baronetcy 2008 'Baronets'

Burke's Peerage and Baronetage 107th Edition, 'Glossary' p. xxxiii

Debrett's Baronetage of England 7th Edtion 'Preface'

'baronet' Encyclopaedia Britannica. Encyclopaedia Britannica Online Library Edition. Encyclopeadia Britannical, 2011. Web. 27 Apr. 2011.

Memoirs of the life of Sir Walter Scott, bart
By John Gibson Lockhart p302,318,331

Sybil Benjamin Disraeli