Reading the Ice and Fire books, I keep on coming across the term Hedge Knight (and, also, the apparently pejorative, Hedge Wizard).
What specifically does the prefix Hedge refer to?
A Hedge Knight is an individual who has been dubbed as a knight, but does not owe his allegiance to any master. I guess Hedge Wizard is about the same.
George R. R. Martin actually uses the term The Hedge Knight for the title of one of his short stories.
I'm not too up on medieval history, but the idea of a roving knight doesn't crop up too much from what I remember - so I think it is something in-world. I certainly haven't seen Hedge Knight used outside of Martin's works.
In general, "Hedge Wizard" refers to either a peasant wizard or a dispossessed apprentice of a wizard who has come to function on his own. In many fantasy settings, it refers to a wizard who is neither a member of the wizard's guild, nor possessed of a license to practice.
A hedge knight is a person, who having been dubbed a knight (elevated to knighthood), but not then sworn to the knighting lord, nor to any of his vassals. The term is reputed to refer to sleeping in the hedges, but as often as not, those hedges marked the boundaries of particular fields or orchards, and could just as easily apply to one camped just beyond, or even living in the peasant villages who worked those fields.
In either case, the term is derogatory in most contexts, as the implication of living in/under the hedges puts one as poorer than a peasant, as even the poorest peasants had a place to build a hut.
A Hedge Knight is a Knight who does not have service with any Lord and does not posses land himself from a Lord.
In Chapter 71 of ADWD, Daenerys recalls following:
Viserys told her tales of knights so poor that they had to sleep beneath the ancient hedges that grew along the byways of the Seven Kingdoms. Dany would have given much and more for a nice thick hedge. Preferably one without an anthill.
As they have no lands or service with a landed Aristocrat, they do not have any housing facility. Therefore they have to roam the seven Kingdoms in search of service (whether permanent or temporary). Because they have no liege lord to pay them and chances of finding temporary services are not very high, these knights are usually low on money and can't afford to live in inns.
So where do they sleep? Outdoors! Specifically, under hedges. Contrary to the apparent impression, Hedges in Westeros are actually very good places to sleep. The older the hedge is, the better it is.
In our world, the prefix "Hedge" would roughly translate to Hobo which most certainly is pejorative. It implies that the said person is landless, jobless, homeless and poor.
Unlike Knighthood, Wizardry is not an established institute in Westeros. Cynical people would mock men claiming to be wizards as frauds while gullible people would believe their claims. If someone is a Hedge Wizard, that means he is also landless, serviceless, poor and homeless, while maintaining that he is a wizard.
Martin often makes slight alterations to historic terminology to make his world unique from ours, for example using "Banner" to refer to a Knight Banneret. The historic "hedge knight" was never referred as such, but, rather, as condottiere (Italian) or compagnies grandes (French). Sir Walter Scott eventually gave them the English name "Freelance" in Ivanhoe. It refers to a knight who is not committed to any particular liege, and is thus free to offer (and often sell) his services.
Martin splits them into two categories; those ennobled and those not ennobled. The former is a hedge knight, while the latter is a free rider.
Hedge wizard, as I've always known it, refers to a self-taught wizard, as opposed to one who apprenticed under another wizard.
A knight in Martin's works refers to a person annointed in the 7 oils, stands vigil in a sept, and takes the knight's vows. The baseborn (concieved out of wedlock) cannot become knights. A hedge knight is the lowest rung on the knightly totem pole, so to speak. They differ from landed knights in that they hold no lands and swear fealty to no lord. However, they may become sworn swords if, during some campaign, a lord wishes to increase his numbers and accepts the hedge knight's oath. After serving as a sworn sword, the knight generally adopts the lifestyle of a hedge knight again.
It would be misleading to think of hedge knights as being similar to ronin. There are some superficial similarities (the gentry hold them in ill regard, and they use their martial prowess to make money), but the best way to gain insight into the meaning of "hedge knight" is to learn about knights, not by comparing them to ronin. I won't go into detail, as your question doesn't concern ronin, but a little research into ronin warriors will give you the info you need to make the distinction yourself.
Think of it this way: It's similar to a Japanese samurai ronin. A hedge knight is essentially a roaming sellsword (they all love to call themselves "Ser So-and-so"), but in Martin's works the lack of common Rule of Law and the depressing lack of honor make the vast majority of hedge knights seem to be only barely above the level of brigands.