She is first seen in a church that Wilf wonders into, babbling to him a story about how the Doctor once killed space satan or something. How does she know this happened? Why is she telling him this? Why is it even relevant? Never explained.
Next she appears on a TV screen telling him that he should help out the Doctor or something, and then she appears to him inside the cactus spaceship telling him about what's at stake and what the Doctor needs to do next. Finally she appears behind Rassilon and makes eye contact with David Tennant, meaning she's someone important or something.
Apparently RTD intended for her to be the Doctor's mother, but then decided to leave it up to the fanbase to decide it. Typical BBC behavior, there.
The problem with her Edit
So why was she such a terrible part of the episode? Well, RTD seems to have a common theme in his storyarcs of having random psychic characters telepathically link with the screenwriter and just suddenly blurt out future information for no real reason other than to entice the viewer to keep watching on. The Woman is pretty much the culmination of this trope, just flat out telling Wilf some exposition to set him on the right track, instead of just... I dunno... writing Wilf so that he gets on the right track by himself? Similarly, during that tediously long scene near the end where Ten is stuck between shooting the Master or Rassilon, she conveniently pops up and uses her eye movements to signal to him that it's the White Point Star he needs to shoot. She might as well have held up the solution written on a flashing fucking neon sign. Why did RTD have to have a random plot convenience character just flat out tell him the answer? Why couldn't the Doctor be smart enough to come up with the conclusion by himself? Seriously, she wasn't necessary at all.
Later appearances Edit
In Hell Bent, there's a woman in the barn playing a clearly motherly role to the Doctor. Is she the same character? She's played by Linda Broughton rather than Claire Bloom. And RTD said he has no idea if The Woman is the Doctor's mum in the first place, while Moffat said he has no idea whether they're the same character, and it's up to the fans to decide.
Maybe the Doctor has two mums. Maybe she's Miranda's daughter Zezanne come back from the future from that one novel before she regenerated into Susan. Maybe Time Lords have totally different family relations than us and she's an older lady from the same Loom. But no, seriously, she's the same character, and the Moff is just trolling in hopes that some fan will be autistic enough to come up with some fanwanky way to make Lungbarrow still canon so he can laugh. (He did miss a chance to make a "half human on my mother's side" reference, however, proving he's not the best troll.)
Still, she actually plays a role, as the Doctor's mum—a simplistic, un-fleshed-out, minor-character role, but a character rather than a random writer's oracle plot device, because Moff != RTD.