|Race||Human (by birth) Dwarf (by upbringing)|
|Age||b. 1969 UC|
|Occupation||Captain in the (City Watch)|
|Physical appearance||6'6" - Very, very muscular, a full head of orange hair|
|Parents||Family (presumably) attacked by bandits near the Copperhead mine. Raised by the king of the mine and his wife.|
|Marital Status||Courting (Angua)|
Men at Arms
Feet of Clay
The Fifth Elephant
The Last Hero
I Shall Wear Midnight
Technically Carrot Ironfoundersson is a dwarf, for he was adopted and raised by dwarfs, even though he is biologically human. This made growing up difficult, especially since dwarfs are smaller than the typical human (Nobby Nobbs, as always is an exception to this rule) and Carrot is taller than the typical human. Carrot's adopted father sent him to Ankh-Morpork to join the Ankh-Morpork City Watch in Guards! Guards!. He is an exemplary policeman, but can be seen by others as too polite. In anyone else it would be suspicious; with Carrot however it is sincere. He has an old but extremely strong and sharp sword, and a funny birthmark.
Carrot is the most consciously and conscientiously law-abiding citizen in Ankh-Morpork. When he arrived, he brought with him the book The Laws and Ordinances of the Cities Ankh and Morpork, a very old book that once belonged to a retired Ankh-Morporkian Watchman. Having been brought up in dwarf culture, Carrot takes everything literally (throughout his appearances, he develops a fuller grasp of figurative language) and believes in obeying rules. The old Watchman's descendant told Carrot about upholding justice, so Carrot enthusiastically enforced the laws as detailed in the old book, the contents of which were unknown to everyone, even old-time Watchmen. This got him into some trouble with the city government, starting with his arresting the head of the Thieves' Guild. Yet, somehow, he managed to bring morale to the Watch just as Lady Sybil Ramkin did. Carrot is now Captain of the combined Ankh-Morpork City Watch; formally he would be called "Captain Ironfoundersson", but most people call him "Captain Carrot".
"Most people" can be taken to mean both "most people who know him" and "most people in Ankh-Morpork". Carrot walks the patrol routes even on his days-off, he talks to everyone and remembers everyone's name, face, family, business, and so on. It is said that he reads the register of taxpayers, for fun. Sometimes a tax-evader makes claims, in Carrot's presence, about having paid his taxes and therefore entitled to the Watch's protection, Carrot will respond by saying that he hasn't seen the name in the taxpayers' register, perhaps the good citizen has been too busy and forgotten to fill out tax forms, and Carrot would bring some along soon. Carrot genuinely thinks everyone, even the boy thugs in the street gangs, is decent and good underneath all the Ankh-Morpork prejudices and violence. For some strange reason, probably what Sgt. Colon calls krisma, people find that they want to meet Carrot's expectations and do what he says. It is a good thing that Carrot wants what's best for everyone, and has far fewer prejudices than most people; otherwise, he might become a ruthless despot forcing his ways on others. He also knows the cultures and languages of many immigrant groups in Ankh-Morpork, and politics and geography of places as far as Überwald.
Carrot is not completely without prejudices, or perhaps quirks. A legacy of his upbringing by Dwarfs, who are reticent on some matters, is that he finds it very, very, difficult to talk about...er... gender-related issues. He therefore found the desire of some Dwarfs to openly differentiate themselves as female to be very difficult to accept, and this prudish, conservative, streak caused one of his first real rows with Angua. Carrot also had red-faced difficulties with the consumer product manufactured by Wallace Sonky. During a conversation with Angua in Men at Arms he reveals that he has a prejudice against the undead, saying "I just don't like them", although he has most likely overcome this discrimination against the "differently alive", seeing as he is now dating one.
Carrot is currently in a relationship with Angua. His previous romantic affairs include the dwarf Minty Rocksmacker back in the mine, and the seamstress Reet, who he rescued from a street thug when newly arrived in Ankh-Morpork.
At the start of Snuff, he has been elevated to Acting-Commander of the Watch so that Sam Vimes can be sent, at crossbow-point, on holiday. Although it isn't necessarily Vetinari who is holding the crossbow.
It is rumoured that Carrot is the last remaining descendant of the kings of Ankh-Morpork, but it has never been voiced too loud. He gets on with people well, he slays dragons, sort of, and is generally liked by a lot of people. Carrot wants no claim to any throne, he likes being in the city watch and Lord Vetinari governing works very well. Occasionally, and in the privacy of the Oblong Office, Carrot does make suggestions that Lord Vetinari finds hard to refuse. Following the events of Men at Arms, it is generally implied that Carrot is actually fully aware of his regal status, but a conversation with Vetinari where he was offered command of the Watch after Vimes's brief retirement included Carrot declining the offer on the grounds that people should obey the rules because they are the rules, and not because Captain Carrot is good at being obeyed. It is obliquely hinted at the end of Jingo that Vimes' elevation to Duke of Ankh had something to do with Carrot's unclaimed status, since only the King, not the Patrician, can create a Duke. He takes advantage of it once more in The Fifth Elephant, noting that the retired, sacked and otherwise deposed men of the watch had sworn an oath to the king, and using this to recall them all before Vimes returned.
Discovered in complete serendipity, I'm really surprised nobody else has found this. Offenbach's 19th century light opera Le roi Carrotte (King Carrot), in which a humble Carrot first becomes human, and then becomes King, deposing a hated tyrant in the process...
See here:- 
According to Pratchett himself, the origins of the name are more mundane: he once had a technician come over to his house with a tall red-haired assistant referred to by the elder professional as "Carrot", and presumably this name stuck. When we are first introduced to Carrot in the books, however, the author takes time to note that young Carrot have gotten his name not because of the hair, "which his father has always clipped short for reasons of Hygiene", but for his broad-shouldered "tapering shape".