Susan is unique in the history of the TV series for two reasons. First, she is the only member of the Doctor’s family to have travelled with him and, secondly, she is already travelling with the Doctor at the beginning of the series. Every other companion, bar none, has some sort of introductory story where they first meet the Doctor before being whisked off through time and space. But not Susan.
Given that the series was keen to preserve a sense of mystery about the Doctor in its early years it’s probably not surprising that we also learn very little about Susan in the time that she spends travelling with the Doctor. In the first episode ‘An Unearthly Child’ we learn that Susan is an alien (hence the title of the episode) albeit one with a good grasp of Earth history (“that isn’t right!” she says on reading a book on the French Revolution) and better understanding of science than any other teenage girl of time.
“I was born in another time, another place, “ she says at one point. While it’s not explicitly stated we can assume that she born in the same place as the Doctor. In the story ‘Marco Polo’ (and there’s no prizes for guessing which historical figure features in that story) she describes her home world as having skes of orange and silver leaves on the trees. This fairly closely resembles how Gallifrey is portrayed on screen in the 1970’s story ‘The Invasion of Time’ and more recently in ‘The Sound of Drums.’
Susan is one assumes a Time Lord (or Time Lady, rather) just as the Doctor is but frustatingly again this is never confirmed, mainly due to the fact that the Doctor isn’t reavealed to be a Time Lord until long after Susan left the series.
There is also some speculation as to whether or not she really is the Doctor’s grand daughter. For example some fan theories suggest that perhaps the Doctor adopted (or perhaps even kidnapped) Susan before starting his travels and that she took to calling him ‘Grandfather’ simply as a sign of affection.
So we are left with is a lot of speculation and little in the way of solid facts; or as Ian Chesterton puts it in the first episode: “Too many questions and not enough answers.” One thing we do know for certain is that Susan came up with the name TARDIS, or so she claims in the first episode. That seems to imply that the name is unique to the Doctor's Ship,similar to naming a sailing ship 'Titanic', although as we see in later years this is not the case. Clearly Susan's name caught on with other Time Lords.
Susan appears to be a reluctant traveller. Although, on the surface she appears to enjoy travelling and exploring every bit as much as her grandfather, deep down she appears to harbour a desire to stop wandering aimlessly and belong somewhere. In the afore mentioned 'Marco Polo' and also in 'The Sensorites' later in the same season she talks wistfully about her home and in the latter story she muses on what it would be like to settle down in one place and time.
This plot point comes to a culmination in the second season's second story: 'The Dalek Invasion of Earth'. Here Susan grows up somewhat as she meets and falls in love with human freedom fighter David Campbell. She's torn between staying on Dalek-ravaged Earth with David or continuing to travel with her grandfather. Seeing her plight the Doctor makes the decision for her and locks her out of the TARDIS.
This action might seem a little harsh but the Doctor is simply doing what he thinks is best. He knows that Susan is not happy travelling any more - indeed, she as good as says that the only reason she is still travelling is because she believes the Doctor still needs her. The Doctor knows that Susan will never leave him voluntarily so he makes the decision to leave her instead.
And that is pretty much the end of Susan's story within the series. Despite saying that one day he'll come back and see her, he never does. Susan does appear again in The Five Doctors, a midd;e-aged woman now, but no mention is made of what has happened to her in the intervening years. Indeed, aside from the First Doctor, none of the other Doctors in that story appear to want to have anything to do with her at all! Perhaps they're just embarassed that they forgot to go and visit her? Sadly this means that we never get to find out if the Doctor ever got to become a great-grandfather.