Fate and the Lady are two gods of the Disc, and like so many gods, they occupy that wishy-washy space between the abstract and not-quite-abstract.
All the gods of Discworld can change any physical aspect about themselves that they choose, except for their eyes. So Fate's eyes are always described as being "but holes opening onto a blackness so remote, so deep that the watcher would feel himself inexorably drawn into the twin pools of infinite night" (2.3.34). In contrast, the Lady's are "bright green, lacking iris or pupil, and they [glow] from within" (2.Prologue.10).
To summarize Fate's character would be kind of redundant. Fate is fate; he's the determiner of all that will happen on the Disc, and his will in the matter is ironclad. As he tells the Lady, no one can cheat him though "everyone tries" (2.3.39). It's a losing battle when it comes to this guy.
The Lady is his opposite: Lady Luck. She is "the Goddess Who Must Not Be Named; those who [seek] her never [find] her, yet she [is] known to come to the aid of those in greatest need" (4.13.48), though she never comes when called by name. As such, she's become a sort of patron deity for Rincewind and Twoflower, blessing them—particularly the former, who is a fan of the sure thing—with an exorbitant amount of luck.
She's also the only one who can match Fate in that godly dice game in the sky. The struggle between these two gods is arguably the main conflict of the entire book. Fate desires outcomes to end as he sees them—that is, with Rincewind dead. The Lady, however, wishes to keep Rincewind alive for whatever random reason she has. Perhaps she's fond of Rincewind. Perhaps she just wants to upset Fate.
She's luck, so who knows? Not us, that's for sure.