What is the proper terminology for a female knight? I hear that it's "dame" or "ser", but that there are specifics beyond that, and I'd like to know more.
There is, sadly, no such thing as a female knight, nor has there ever been, to the best of my knowledge.
Women who are part of chivalric orders, certainly in the British honours system, are not titled knights, but usually either 'lady' or 'dame' (depending on the order); this is pretty much a twentieth century thing, as women were generally excluded from such orders in earlier centuries.
If you're wondering about actual knights (in armour) there's even less to go on. Knights were professional aristocratic soldiers, and women weren't soldiers. There was no title for knights in the central middle ages (they tend to be identified as 'John of Townsville, knight'); from the fourteenth century you start to see the classic 'sir' being applied in English.
'Ser' is a fantasy word invented by George R.R. Martin and is used only in Westeros.