A
title of
nobility is held by a single person at any one time. If Alfred is the Duke of
Rutland, then his son Bertram is
not the
Duke of Rutland, nor the
Marquess of Rutland, nor the anything else. Bertram is a
commoner. His father is a
nobleman, with a
peerage, but Bertram hasn't got a peerage, but will
inherit his father's when his father dies.*
The children of lords do not have actual titles in their own right, but they have a distinctive style, i.e. how you refer to them. Daughters and younger sons are styled Lord or Lady or the Honourable (the exact rules depend on whether they're children of Dukes, or of Marquesses, or whatever). The family surname of the Dukes of Rutland is Manners: so a younger son would be styled e.g. Lord Charles Manners.
The eldest son of a lord, i.e. his heir apparent, has a different style. The eldest son uses a courtesy title. He uses a title (not a style) which is not his. He is styled by one of his father's titles.
A lord will often have more than one title. For example, The Duke of Kent is also The Earl of St Andrews and The Baron Downpatrick. The same man holds all three titles. He is referred to by his highest-ranking one, Duke of Kent. Now his son is a commoner. But as a courtesy, his son, instead of merely being Lord Edward Windsor (or whatever his name is) is styled with his father's second title, Earl of St Andrews.
The person commonly referred to as Earl of St Andrews is not The Earl of St Andrews. (The 'the' is significant, but the significance is usually lost.) Moreover, Earl of St Andrews has a son and heir, and this young lad is styled with his grandfather's third title, Baron Downpatrick.
This only applies to sons (and grandsons). Any other heir presumptive to a title - that is, the person who's currently next in line to get it - has no special style.
The right of hereditary peers to sit in the House of Lords was recently abolished, but until then the father could sit in the Lords and was therefore not eligible to stand for election to the House of Commons. The sons and grandsons, being commoners, could stand for the Commons.
However by a peculiar quirk it was possible, and very occasionally done, for the Crown to issue a writ in acceleration, calling a son to sit in the Lords, using a minor title of his father's, while the father was still alive. It appears that the son's courtesy title was not legally transformed into his actual title, which still remained with the father.
Strictly speaking, wives' titles are courtesy titles too: a woman can inherit a small number of the titles, such as Countess of Mar, but normally a duchess is so called because she's the wife of a duke, and not strictly a duchess herself. So too are 'Lord' and 'the Honourable' in front of other sons' and daughters' names. (See under Lord for who gets to be called what). But usually 'courtesy title' refers to the habit of sons being styled by minor titles of their fathers.
* Bertram could hold a title of his own if he'd inherited one from his mother.
For advanced students only
A remarkable example has come to my attention. The
heir apparent of the present
Duke of Buccleuch and Queensberry is styled by courtesy
Earl of Dalkeith. Yet these are the relevant titles the Duke holds, with their dates of creation: Earl of
Buccleuch 1619, Earl of
Doncaster 1662 (English), Duke of Buccleuch 1663, Earl of
Dalkeith 1663, Duke of
Queensberry 1684, Marquis of
Dumfriesshire 1684. There is a marquisate and two more senior earldoms: why isn't his son called one of those?
I don't know, but my theory is that the son gets the second title of his father's senior title. The dukedom of Buccleuch is older (1663) than that of Queensberry (1684), so only those titles that come with the Buccleuch line are used for courtesy titles. (The two lines were united by inheritance in about 1800, as I recall, though the above entry in Who's Who doesn't mention it.) From the dates it looks as if the Dumfriesshire title was created for the Queensberry one.
He can't be styled Earl of Buccleuch because you'd in effect have two Lords Buccleuch. That leaves the English title of Doncaster. Now the Duke of Buccleuch was obviously Earl of Buccleuch before his elevation: who was the Duke of Queensberry before his? Almost certainly Earl of Doncaster: he's unlikely to have been promoted straight to duke from a lower title. So on this reasoning the earldom of Doncaster is inherited in the junior (Queensberry) line, so is not available as a courtesy title in the senior (Buccleuch) line, any more than the marquisate is.