"While evil flourishes and wrongs grow rank, while men are persecuted and women wronged, while weak things, human or animal, are maltreated, there is no rest for me beneath the skies, nor peace at any board or bed. Farewell!"
They seek out wrongs to right, generally on a small scale — a town beset by bandits, a dragon preying on the locals, etc. Only occasionally will they be pitted against an Evil Overlord with earth-conquering ambitions. They often have one specific quest they are on, but you can count on them running into unrelated trouble along the way.
Sometimes, they are The Stoic, almost to the point of being less a character and more a physical incarnation of justice. They are very prone to having a Mysterious Past. This type generally overlaps with The Drifter.
Rurouni Kenshin did this for ten years prior to the series, which starts with him temporarily suspending his Walking the Earth to stay in a dojo in Tokyo. His variation is that unless he's actively kicking butt, his looks, natural personality, and Obfuscating Stupidity work together to make sure nobody takes him seriously.
The anime version also wears pink. (Based on color spreads, manga Kenshin has a red haori and a blue haori. Where he carried the one he wasn't currently wearing in his traveling days is a question for the ages, since he had no baggage.)
Vash the Stampede does this, too, Obfuscating Stupidity turned Up to Eleven. He's a little less formal about it than Kenshin, because he hasn't got a vow, this is just his lifestyle, and he only learned to fight out of necessity, so he doesn't have quite the same type of warrior ethos as a proper knight or samurai.
The Elric brothers seem like this at the beginning of Fullmetal Alchemist, before the wider-reaching plot arcs start up—they travel from place to place, righting wrongs and searching for leads on how to get their bodies back.
Especially true in the 2003 anime version, which had a few extra filler episodes near the beginning that consisted almost entirely of the Elrics going somewhere, righting some wrong, and going along their merry way, with the episode tying in minimally or not at all to the long-term plot.
Parn from Record Of Lodoss War. He is known as the Free Knight because he officially holds loyalty to no kingdom and helps anyone in need, no matter their allegiance or nationality.
The titular character of Lucky Luke embodies this, to the point where the last panel of every album is him riding off into the sunset, singing I'm a lonesome cowboy.
Dark Horse and Star Wars are coming out with a comic book series called Knight Errant. What's known is that it will have a lone Jedi Knight during the dark time before the final battle of Ruusan, where the Sith outnumber the Jedi, and the Republic has been whittled down to just the areas around the core. The lone Jedi will be going around to planets in the Sith zone and fighting the good fight.
DCU's "The Warlord" seems to be based on this trope. Travis Morgan, aka "The Warlord", was a USAF lieutenant colonel who crash landed in Skartaris, a world inside the hollow Earth. A modern man injected into a world of sword & sorcery, he falls in love with and marries Tara, the warrior-queen of Shambhala. The character avoids a bad cliché by not becoming Skartaris’ leader and attempting to impose his values on it, but neither can he sit idly by at the royal court whilst knowing how much is 'wrong' in the rest of the world. As a result, he is constantly absent- journeying around Skartaris as a knight errant, enjoying the role of an adventurer far more than that of a king.
Invoked by Gilbert in The Sandman, who insists upon being Rose Walker's "Knight Errant" in her search for her brother.
The Jedi from Star Wars often act as Knights Errant. A New Hope itself was adapted from The Hidden Fortress, a samurai movie. By the book Obi-Wan is a good specific example because he is an ideal Jedi Knight.
Wuxia novels, films, plays, etc. are full of such figures, and the term even translates to something like "wandering knight." This derives from Chinese knighthood, which isn't a widely-known subject in the West.
The protagonist of The Last Knight is a knight errant who lives about 200 years after errantry has gone out of fashion. His squire is a former con man who at first comes along mostly because it keeps him out of jail, and later to look after his employer, who's prone to Honor Before Reason.
The Dark Tower: Roland Deschain is a Gunslinger from another dimension who is descended from Arthur Eld and wields guns that were forged from Excalibur. He's on an epic quest to reach the Dark Tower and fight the forces of evil. The story is very loosely based on the poem "Childe Roland to the Dark Tower Came," which is itself based on The Song Of Roland.
Colonel Next from Thursday Next is consistently described as a "time-travelling Knight Errant", usually turning up in the nick of time to save the world.
Malik ibn Ibrahim from the ebook anthology Wandering Djinn seems to fall into this category constantly.
The Knights of the Cross in The Dresden Files fit this archetype; even Michael, who has a home, a wife, and a small army of children spends much of his time traveling the globe, righting wrongs and fighting evil. Their main job, as it turns out, is somewhat different - they're actually supposed to get the Denarians' hosts to repent and give up the Denarians. The doing random good seems to only be their secondary function.
Alana from the Tortall Universe starts out in The Woman Who Rides Like a Man to escape court and find adventure.
Miles Vorkosigan has been diagnosed as having Knight Errant tendencies - by his mother.
Simon Templar, more commonly known as The Saint, is this in many of creator Leslie Charteris's stories (one story collection was even titled Saint Errant) and in some episodes of the TV series that starred Roger Moore. Not so much in the Val Kilmer movie.
John Shannow, the eponymous Jerusalem Man in David Gemmell's series of that name, starts by wandering a post-Apocalyptic Earth in a vague search for Jerusalem, but is repeatedly distracted by evil.
The first novel in the Mech Warrior: Dark Age series centers around a Knight Errant, essentially a Mech Warrior who had the job of traveling to trouble spots around the galaxy and basically lying low to gather information and take action in case something came up.
Of course, this wasn't clear for the first several chapters, where he seemed to be a blue-collar type with a Mysterious Past who gets caught up with a political extremist group. The Reveal that he's a knight happens around the same time that the female mechwarrior who just arrested him and took him aboard her ship for questioning plants a kiss on him, putting everything that occured before that in a new light.
In Patricia A. McKillip's "The Kelpie", Nice Guy Ned is suggested for this in a painting; he wishes he could be evil for once, and is asked if he can settle for triumphant.
The Justicar, hero of Paul Kidd's trilogy of Greyhawk novels White Plume Mountain, Descent Into The Depths Of The Earth, and Queen Of The Demonweb Pits is made of this trope. He does everything but travel alone. Much to his discomfort, he accumulates ever more companions through the series. He starts with just a sentient hellhound skin, and collects a faerie princess who is also an accomplished mage, a drunken human teamster (who later becomes a drunken badger), a female sphynx, a young male human soldier, a talking sword, and a non-evil demon (who just wants a nice clean, calm, safe place to settle down with her boyfriend).
The main protagonist of The Dragon by E. Schwartz. He is also called Lancelot.
The A-Team, similarly: the knight-errant needs to be solitary.
Angel Parodied when Wesley attempts to be this when first introduced.
In Doctor Who, the Doctor often fits this trope, even if he's not actively seeking wrongs to right, he seldom hesitates to get involved when he runs into one. He's had many companions, but they act more like "squires" than true partners and none of them stay with him forever. His sixth incarnation outright calls himself a Knight Errant at least once.
Sam and Dean in Supernatural fit this trope, especially the former. While they have some other overarching goals each season - finding their father in Season One, trying to negate Dean's contract in Season Three, averting the Apocalypse in Season Five, fighting the Leviathans in Season Seven - they always stop by in whatever wayward towns are being haunted even if they don't have personal reasons to, and deal with the supernatural threats there. Although they ARE involved in criminal activity - false credit cards, fake IDs - this is done for sustenance and to further their goals each time, as they don't get paid for their work. Other Hunters probably count as well - particularly John Winchester before his death, Bobby Singer, and Ellen & Jo Harvelle as of the Season Two finale.
Seems like pretty much everyone in their line of work that doesn't stay in one fixed location (i.e. most of them, as hunters tend to leave messes behind, and there are only so many monsters in one location) have to engage of some level of criminal activity. Frequent ones are grave desecration and breaking and entering, but on the Headscratchers page for the series, it's been mentioned that the boys probably at least loot the corpses of those killed by the monsters they hunt. Gas isn't cheap.
RPG adventuring parties aren't normally an example of this; a key part of this trope is that the knight is solitary. "Solo" campaigns, with a single player and a DM, are not uncommon however.
Bretonnia, one of the factions in Warhammer Fantasy Battles, has a unit type called Knights Errant, representing the young, impetuous knights whose courage sometimes outpaces their skill. Their special rules reflect these details. A better example of this trope, Questing Knights, are knights who have given up their feudal responsibilities to quest for the Grail and the Lady of the Lake; their oaths command them never to stay in one place for too long, and they keep righting wrongs until they die or drink from the Grail.
Magic: The Gathering has Gideon Jura. Elspeth Tirel as well, who actually bears the title of Knight-Errant. Elspeth's defining trait is that she doesn't want to planeswalk; she just wants to find paradise and stay there.
Of course, this is Magic. Her first home was destroyed by Phyrexia, and her adopted home of Bant became culturally contaminated when the shards of Alara merged. Now she's trying to take the fight to Phyrexia, and the outlook is very bad.
And then of course there is the vanilla card from even further back Knight Errant.
Samara, an Asari party member in Mass Effect 2 is a Justicar. According to her "the closest human equivalent would be a knight errant, with perhaps a bit of samurai".
Paragon Shepard can spend most of their time travelling the galaxy and selflessly going out of the way to put right any wrong they encounter.
Sanger Zonvolt from the Super Robot Wars series, the Sword That Smites Evil who will show up whenever evil arises. And will do anything to protect his girl, namely Sophia Nate.
Adol Christin of Ys decided when he was a kid that he'd travel the world, righting wrongs and nailing chicks. Well, the second part's mostly inferred.
Kingdom Hearts gave us Sora, who constantly travels from world to world, never stopping or staying very long, righting whatever wrongs he happens to come across along the way, from finding lost dogs to toppling local tyrants. He might have a few too many friends to count as a truly classical example, but 90% of them are the "leave them behind when you leave town" variety. By Kingdom Hearts 3D [Dream Drop Distance], it's earned him an official knighting by then-Princess Minnie.