There are several ways to answer this question of course. We could position ourselves outside the logic of representation and say ‘s/he's a character in a long-running BBC TV science-fiction drama serial’. Or we could step inside the logic of representation and say ‘s/he is a centuries-old humanoid alien, a so-called Time Lord from the planet Gallifrey, who can travel through time and space’. Other answers might include: popular, often recast, British, a lucrative piece of intellectual property and so on. But there's a particular answer to this question that is, I think, crucial to understanding the character's enduring appeal, one broadly (I think) overlooked by fans and critics. Doctor Who, as he initially appeared, and through most, but not—as I'll discuss in a moment—all, of his incarnations, is a gentleman.
I'm going to talk about the Doctor as a man for the next few paragraphs, since I'm trying to address what the character has been so far rather than guess what the franchise's future holds. And although it is wonderful to see a woman play the role, it can't be denied that questions of masculinity have always been intricately bound up with discourses of class and gentility in the English social and ideological imaginarium. So what is the Doctor? He is a man of breeding and wealth (the two things don't always go together, but this case they do), factors that enable him to evade the responsibilities of work that bear down upon the rest of us. Unoppressed by such necessity the Doctor can do what he likes, and what he likes is, broadly: travel and a kind of elaborate philanthrophy, both perfectly respectable gentlemanly pursuits. The Doctor has tremendous charm and is at ease in whatever company he finds himself; he has immaculate manners and solicitude for others—Robin Gilmour's The Idea of the Gentleman in the Victorian Novel (1982) defines a gentleman as, amongst other things, somebody who never knowingly causes pain to another. And I might cite Gilmour's book to raise two other features of the Doctor's personality that are not only gentlemanly but more specifically Victorian- or Edwardian-gentlemanly: he is eccentric, and he is not a snob.
Eccentricity is a marker of class in the practical sense that a gentleman can get away with acting oddly and indulging his personal crotchets in a way that would lead to a working man (or woman) losing their jobs, or being otherwise socially sanctioned. The gentleman can do as the common man cannot and dress strangely, behave peculiarly, ignore social convention, lark and sport. And the figure of the gentleman as eccentric is increasingly a feature of later Victorian and Edwardian representations of (celebrations of, we might say) the type. So Richardson's Sir Charles Grandison is impeccably conventional in his behaviour; but by the time we get to, say, Sherlock Holmes his whimsies of manner and habit, from playing the violin badly to keeping his cigars in the coal-scuttle and his tobacco in the toe end of a Persian slipper (and, indeed, taking cocaine) are absolutely part of his gentility.
As for disdaining snobbery—which we see in the way the Doctor goes out of his way to be polite to nobodies, to folk working menial jobs, or without any kind of job, or people otherwise manifesting generic Cribbinsness—well: this becomes quite specifically part of the armoury of gentleman in the second half of the nineteenth-century, and does so because one writer, Thackeray (who was absolutely fascinated with the figure of the gentleman) made it so. Thackeray's Book of Snobs (1848) is little read nowadays, but I'd make the argument that it has proved more influential and enduring than Thackeray's most famous novel Vanity Fair, coincidentally published the same year. ‘Let us remember,’ P N Furbank says,
that it was Thackeray who invented our current concepts ‘snob’ and ‘snobbery’, ones which no other country possesses in quite that form. I mean the form of an infinite regress, by which the abusive term ‘snob’, in the sense of low, vulgar, cobbler-like, is not rejected but is transferred to, or reversed upon, those who would use it in that sense. Thus, the vulgarest thing (sense 2) you can do is to look down on somebody for being vulgar (sense 1) or to curry favour with somebody for being non-vulgar.This is important, because a large part of what for want of a better word I'm going to call the Doctor's ‘progressive’ reputation depends upon his repudiation of snobbery. It's something at once a marker of class and, as Furbank notes, of Englishness, and both those things feed into the representation of the Doctor.
Or largely do, because of course there have been incarnations of the character as not English (Capaldi's Scottish 12th Doctor) and not well-bred (Eccleston's working-class 9th Doctor). But of these the interesting one is Eccleston's, since Capaldi, by playing the Doctor as an elderly eccentric Scottish gentleman, was hardly moving the figure far from the way Hartnell, Troughton, Pertwee and Hurt played the role as elderly eccentric English gentleman, or indeed the way Baker, Davison, Baker mk 2, McCoy [edit: commentators below point out that McCoy's Doctor was actually Scottish], McGann, Tennant and Smith played the role as a youthful eccentric English gentleman. The standout is Eccleston, a popular iteration of the Doctor amongst Whovians and a very fine actor, but somebody who proved never at ease in the role, who left the show prematurely after disagreements with the production team (and who has, uniquely amongst living former Who-actors, never returned to it). The most notable through-line in Eccleston's ninth-doctor adventures was the introduction of a working-class ‘companion’, Billie Piper's Rose Tyler, although the romance between Rose and the Doctor was postponed until Eccleston's one-series tenure ended and the prettier and posher David Tennant took up the role (Tennant, a Scot, affected an RP accent for his portrayal). It's a little hard to assess the place, or significance, of the Eccleston Doctor in the larger Whocanon, I think: I don't know if it would be right to say that it represented the Doctor in effect ‘slumming it’, and it would be lazy to present Eccleston as the exception that proved my larger rule.
Still, the main point I'm trying to make here is that the positive valences of ‘the gentleman’ absolutely cannot be separated out from the larger ideological structures of class and privilege in which they occur and which, in various ways, they sustain. There is a lot of (as it were) semiotic leverage applied to the concept that attempts to distance it from questions of class, to suggest that one can be a gentleman without breeding or money if one behaves in a certain way; but actually this is all part of a larger false consciousness. Irrespective of all that other touchy-feely stuff about having good manners and not being rude to one's social inferiors, the gentleman is a figure with a structurally copestoning role in social hierarchy. Above I quoted Robin Gilmour (who, incidentally, taught me Victorian Literature at Aberdeen University in the 1980s, and is one reason why I specialised in that period as an academic) to the effect that a gentleman never knowingly causes another person pain. But whilst Gilmour does say this, the larger thesis of The Idea of the Gentleman in the Victorian Novel is that the Victorian preoccupation with the gentleman manifested an underlying social fact, ‘the transference of ruling power from the landowning and aristocratic élite to the rising middle classes’. The gentleman is a different sort of social superior to the older aristocrat (who was, by and large, much less concerned with the wellbeing of his inferiors, and more defined by a haughty insistence on his own honour and privilege), and he came about, partly, in reaction to the Revolutionary culling of aristocracy. But the gentleman remains a type of social superiority for all that.
And the British, and more specifically English, obsession with the figure of the gentleman is all tangled up with a larger socially reactionary anti-egalitarianism. For most of its history the fundamental logic of the British Army has been that proletarian men should be led into battle by officers whose most important qualification has not been leadership skills or grasp of military strategy, but simply breeding—it's insane, really, but there you are. Sherlock Holmes is, as has been mentioned, a gentleman. Old Etonians James Bond and Captain Hook are both gentlemen. The TV decade that gave us Doctor Who also gave us the now forgotten (but in-its-day huge) Adam Adamant, a gentleman, and the still very much remembered The Prisoner, in which Patrick McGoohan plays a gentleman spy whose sang froid remains untouched through all manner of weirdnesses—as well as the gentlemanly Steed in The Avengers. Quatermass is a gentleman. Simon Templar is a gentleman.
Today, when the limited but significant gains made in the direction of practical social egalitarianism of the 1960s and 1970s are being so comprehensively rolled back all around us, the British obsession with the gentleman has experienced something of a reflorescence. Our film and TV stars are all Cumberbatchy, Redmaynesque, Hiddlestonian public school posh-boys; our hits are Downton Abbeys and endless adaptations of Jane Austen, our movies are gentlehobbits defeating the forces of evil with the power of their politeness and persistence, or fantasy versions of Eton churning out gentlewizards of both sexes (if you think Harry Potter is about anything other than a uniquely English version of class you're fooling yourself: it's about finding the right line between the cruel and outmoded aristocratic values of Voldemort and his clan on the one hand, and the vulgar money-grubbing of the goblins and tabloid press and suburban lower-middle-class ghastliness on the other). Roald Dahl's enduringly popular fictions combine the sadism of James Bond with the eccentricity of the archetypal English gentleman. Much British TV comedy depends upon audiences intuitively grasping the unspoken absurdities of class, which perhaps explains why some of the shows that have enjoyed the biggest domestic success have not travelled well abroad. For example the central, as-it-were structural ‘gag’ of Dad's Army is that Captain Mainwaring has a higher military rank but a lower social rank than the gentleman Sergeant Wilson. Out of this mismatch whole series of hilarity are construed.
Still, soon we will have a female Doctor and that will be interesting. Because the semiology of the gentlewoman differs in important ways to the semiology of the gentleman. The latter exists as a means of fixing class hierarchy by redefining that structure's apex, and therefore exists as a way of resisting any levelling modes social change: the Doctor might be friendly, but he's always going to (literally) have more heart than we do. But the structural imbalance of gender means that the gentlewoman figures rather differently. This has to do with what has historically always been true: that as women broadly have less power and social prestige than men, so gentlewomen have less power and social prestige than gentlemen. It also reflects the social facts of marriage: a man is born into his status as a gentleman, but a woman might marry into it—one of the ways that Richardson's Mr B. attempts to seduce Pamela is by making the rather threatening promise ‘I will make a Gentlewoman of you.’ Pamela isn't taken in, partly because she shares with her class and time the gut-feeling that women so elevated aren't really gentlewomen: so she scoffs in one of her letters that ‘this very Gentleman (yes, I must call him Gentleman, tho’ he has fallen from the Merit of that Title) has degraded himself to offer Freedoms to his poor Servant!’ Defoe’s Moll Flanders ruefully recalls her youthful dreams of social elevation, when she dreamed of being a ‘Gentlewoman’ and ‘sit[ting] at my window dressed in fine clothes’. Of course some women are born into the requisite social class, but although they are permitted some eccentricities of manner these ought not to trespass so far as a gentleman's for fear of ruining their chances of making a good marriage and sinking into the risible-tragic status of a batty old maid. But the emphasis in the polysyllable gentleman is on the man; gentlewomen are subsidiary figures. Through the 19th century often manic preoccupation with the figure of the gentleman married to an equally desperate engagement with questions of manliness. The period was marked by the deeply homosocial nature of middle and upper-class English cultural life, from same-sex schooling to same-sex gentleman's clubs and pastimes, the sexes being highly segregated. Which means that situating a woman, like Jodie Whittaker, in this particular position is a much more radical act than is sometimes realised.
All this is a roundabout way of saying: I have some specific hopes for the new female Who. It certainly seems to me that many of the reactionaries who greeted the news of her casting with howls of outrage were only partly motivated by misogyny (though of course they were motivated by that); they were trying, with the hysterical volume of their complaining, to rally to the defence of class itself as a defining feature of British self-identity. It will be interesting to see how Whittaker and her scriptwriters take the character, and to what extent the role can be reconfigured to escape this particular straitjacket.