Hedge wizard
The term hedge wizard comes from "hedge witch", a term used in folklore for the cunning folk or local herb-doctors who also use spells and charms to heal the sick.
In fantasy literature, a hedge wizard or hedge magician is generally a wizard of low ability, usually self-taught or with a low education background as opposed to the common examples of being apprenticed to a mentor or studying through a structured educational system. Some fictional backgrounds identify them more with rural than urban backgrounds. In the novels of Mercedes Lackey, the term is derogatory, describing a character as incompetent, uneducated, of lower social standing or of lesser power. It is similar to calling someone a hack writer or a slob but specific to practitioners of magic in these stories.
In role-playing games and video games a hedge wizard is usually a weaker wizard encountered when still at a lower level. With a limited number of spells and lacking in power they are normally easy to defeat. In the Ars Magica roleplaying game, a hedge wizard is any mage not of the Order of Hermes, who claims the largest monopoly on power.
Discworld
In Terry Pratchett's Discworld series of novels, the term 'hedge wizard' is used to describe a wizard who specializes in the magical properties of plants. Although this does fit with the idea of a more rural wizard, given Pratchett's humorous proclivities it is likely here to be a deliberate pun on the word 'hedge'. For example, it is narrated: "If you invited a hedge wizard to a party he would spend half the evening talking to your potted plant. And he would spend the other half listening [to it]."
The Magicians
In the new Syfy series, based upon Lev Grossman's novel, the term 'hedge witch' is used to describe people who were denied entry into the Brakebills college of magic (who promptly have their memory wiped), or kicked out due to rule-breaking in later years (who instead of having their memory wiped, have all of their magical ability and knowledge stripped from them), they can do little more than party tricks due to lacking guidance and resources.