Page 6. " representing a borough in three successive parliaments, exertions of loyalty, and dignity of baronet, in the first year of Charles II. "

Sir Walter Elliot is a baronet.  In the English nobility, a baronet is usually higher in status than a knight of the realm, but lower than a viscount or a baron.  Lady Russell is the widow of a knight, so, in order of precedence, she is lower than Sir Walter who is a baronet in his own right.

Sir Walter's title is not sufficiently powerful to enable him to pass on courtesy titles to his female children (notice the Viscountess Dalrymple's daughter has the title 'Honourable' in front of her name and Elizabeth is 'Miss' Elliot).

Page 9. " For they must have been seen together, he observed, once at Tattersal's, and twice in the lobby of the House of Commons. "

 

Tattersalls, founded by Richard Tattersall in the mid-19th century, was a horse auctioneers at Hyde Park Corner in 1814.  Patronized by the upper classes, it was a place for sportsmen to converge, converse, and engage in some gambling.

The establishment has long since relocated, but it is still very much in existence. 

Page 20. " the most deplorable looking personage you can imagine, his face the colour of mahogany "

 

Mahogany
Creative Commons Attribution Share AlikeMahogany - Credit: J.M. Garg

 

Sir Walter Elliot describes the appearance of an admiral he bumped into.  As is usual with Sir Walter, the admiral's looks are the primary means by which he judges the man. In contrast to today's standards, tanned skin was then considered unattractive.

Here, in comparing him to mahogany, a wood ages by darkening, he is showing the current interest in things exotic.  The mahogany he refers to most likely came from the West Indies, or the Bahamas.

Page 22. " Admiral Croft was a native of Somersetshire, who having acquired a very handsome fortune, was wishing to settle in his own country, and had come down to Taunton in order to look at some advertised places in that immediate neighbourhood "

Taunton
Creative Commons Attribution Share AlikeTaunton - Credit: unknown, wikimedia
 Taunton is the county town of Somerset, and its most populated city. It gets its name from the nearby Tone river. This would have been a central place to asks questions and gain information about the county in Sir Walter's day. 

Google Map

 

Page 22. " He is rear admiral of the white. He was in the Trafalgar action, and has been in the East Indies since; "
Trafalgar Battle
Public DomainTrafalgar Battle - Credit: Nicholas Pocock

 

Thus Anne Elliot describes Admiral Croft, the brother-in-law of her erstwhile fiance, to her father, who remains ignorant of most things naval.

Rear admiral is the lowest designation of the highest office in the Royal Navy (admiral).  The designation rear admiral of the white means that he is not in the lowest rank of rear admiral, but the middle.

The Trafalgar Action refers to the Battle of Trafalgar against Napoleon in 1805, where he would have been under the command of Horatio Nelson, who reached the rank of Vice Admiral of the White. 

The Nelson Society

Some of his letters, etc.

 

The East Indies, a long way from the West Indies where Captain Wentworth had such luck with the Asp, were made up of the islands of what is now referred to as Southeast Asia, including the Philippines and Singapore.

 

Map of the East Indies
Public DomainMap of the East Indies - Credit: William Dampier