A Mini-Boss, Sub Boss or Mid Boss is a distinct, generally unique, stronger-than-average monster that you encounter usually halfway to two-thirds through the level/dungeon/etc. It is noteworthy because it's tougher than any ordinary enemy (and isn't encountered under normal conditions like a Giant Mook), yet it still isn't as strong as the actual boss that awaits you at the end. In story terms, the Mini-Boss is often The Dragon to the level boss.
Some games would have worlds split up into levels, and the level bosses would be mid bosses with the world boss being the "proper" boss.
Recurring antagonists, such as the Goldfish Poop Gang and the Quirky Miniboss Squad, are often mini-bosses.
In the days when Nintendo Hard was the norm, this was especially sadistic. But as games got easier, such enemies were often just a little harder than the regular Mooks, and in some cases would just be Breather Bosses. But it's not always the case. Some lucky ones might even be a Wake Up Call Boss or That One Boss.
Fighting games like to use the "New Challenger" screen normally used for when a second player joins when a midboss arrives.
If there is one, the reward for defeating the Mini-Boss is usually a map of the level, the featured item or weapon of the dungeon (as in the case of The Legend of Zelda games), or a Plot Coupon, such as the Boss Key.
May return as a regular enemy later in the game. Of course, normal bosses may become sub-bosses later as well.
Compare Smash Mook (and some often are), Elite Mooks and Disc One Final Boss.
Contrast Final Boss, Boss in Mook Clothing.
Street Fighter Alpha 3 had you fight a character relevant to your own character's story halfway through the game; this has shown up in other Fighting Games and are often referred to as "story battles".
This returns in Street Fighter IV, happening just before the final battle in each character's story mode.
God Hand loves these; every stage has at least one, and most have two or better.
In Disgaea, this is spoofed with Recurring BossVyers, known to himself as "The Dark Adonis", gets stuck with the nickname "Mid-Boss". This is the characters calling him a minor obstacle, not an actual description of his effects on the game or story. (He's actually the final boss of three of the four chapters in which he is fought.)
Legend of Dragoon is a rare instance of the sub bosses are often as powerful, if not more than the normal bosses.
The two bosses you fight in the volcano during disc one come to mind: Virage and the Flame Bird. The Virage is fought part-way through the level, and serves to prove why they were so feared in ages past. Then you fight the Flame Bird (which most people had forgotten about, after the trauma of the Virage battle), and it turns out to have twice as much HP, but only half the fight.
Mega Man 8 also had a miniboss at the halfway point of half the levels (Sword Man, Clown Man, Aqua Man and Grenade Man).
Mega Man X series: Usually 4 of the eight stages would have at least one.
Inverted in two stages of Mega Man ZX Advent, when the main boss is at the beginning of one level and the midboss of that level is at the end.
At least since Kirby's Adventure, Kirby games have had a bunch of recurring ones. In the games with a Boss Rush mode, they appear again in groups to make up for the lack of power compared to a normal boss. Some of them provide hard to come by abilities such as Cook.
The series as a whole also has Kracko Jr., which is an easier version of Kracko, a boss (and is usually fought in the same level).
The Darius series has Sub Bosses as tradition. Particularly notable are the Sub Bosses of Darius Gaiden; each sub-boss has a spherical orb, usually on the top of it; if you destroy just that part, you can collect the orb, causing the sub-boss to pull a Heel Face Turn and fight for you! Though, it slowly explodes over time and eventually dies. For those who play this game for score, clearing the game nets a huge bonus for each sub-boss captured.
In G-Darius, your ship has the ability to capture all regular enemies enemies and make them fight for you, including Mini-Bosses, although the difference this time around is that you need to first shoot off the gold-colored shielding with normal shots before they can be captured. Once captured, they stick around until they take too much hits from the other enemies, and each one has a special attack they used against you, useable by inputting a set of joystick motions, much like in Fighting Games. And yes, like in the previous game, keeping them alive until to the end of the stage is worth a large score bonus, although they can be utilized to cause a long-lasting Smart Bomb explosion or a stronger-than-normal Wave Motion Gun blast.
Star Fox, Star Fox 64, and Star Fox Assault sometimes have a stronger enemy appear about halfway through the level, although you don't have to defeat them to progress. A straight example would be the Venom Guardians' second fights (The first fights are full fledged bosses) in the original: Phantron, the Galactic Riders, and the Great Commander, but which you fight depends on which route you pice. In Star Fox 64, either Star Wolf or the Golemech on either Venom Route qualify as this. Command also has some minibosses guarding motherships in the harder levels.
Star Wolf collectively act as a full-fledged boss fight, at least in 64.
The Ace Combat games usually have this in the form of either one-time-appearance enemies (such as post-mission update enemies) or the antagonist ace squadrons, such as Yellow Squadron and Strigon Team; the former becomes a Degraded Boss by 04's final mission though.
The Touhou series is very fond of midbosses. With the exception of (as of Ten Desires) two stages, every stage has one. Sometimes more, though this is about as rare as not having one. The thing is, every boss has to have a unique character design and profile, so dedicated midbosses are rare. Which means that it's usually the same character as the stage's actual boss, even if this makes no sense from a story perspective. Other times you get stage bosses midbossing for other characters (sometimes between games!), though this is usually explained.
To drive home how much Touhou likes this trope, Phantasmagoria of Flower View has a recurring midboss. Phantasmagoria of Flower View is a versus shooter, and as such doesn't have stages. Said character, Lily White, is also a midboss in all other games where she appears, outside endings.
Also, in many games there's one fairy somewhere who looks like every other fairy, has no name, but is very tough and powerful by fairy standards.
Metroid games often have mini bosses. In Super Metroid and Metroid Zero Mission all bosses other than final ones and the ones you need to kill to open a way to the final boss are considered minibosses. In Metroid Prime minibosses and actual bosses are easily distinquished: minibosses don't have a health bar and tend to become recurring enemies later on. In Echoes there is no clear distinction as all bosses have a healthbar, but the Energy Controller guardians are often considered to be main bosses and the item guardians minibosses (although in this case some of the most annoying fights are item guardians). Corruption has several minibosses that you fight about halfway through the zones and planets (the actual bosses are the Leviathan Guardians).
Interestingly, the very first game referred to the two required-to-enter-the-final-area bosses as "mini-bosses", even though they were basically full-fledged bosses. Also interestingly, Metroid II pretty much has no major bosses at all; it only has mini-bosses (Metroid evolutions of increasing strength) and a final boss.
Sonic 3 & Knuckles features two levels (Acts) per thematic area (Zone). In earlier games, there would only be a boss at the end of the zone, but in S3&K, there is also a sub-boss at the end of each first act. They are distinct from other bosses, in that they are autonomous, not controlled by Eggman/Eggrobo.
An interesting thing about the Sonic 3 mini-bosses is that if the game is locked on to Sonic & Knuckles, they use the mini-boss music from S&K instead of Sonic 3's mini-boss music. A Good Bad Bug reveals that the S&K mini-boss music is actually on the Sonic 3 cartridge.
Super Mario 64 has the King Mook enemies that don't have a background boss theme (Big Boo, the Big Bullies, Mr. I, etc.), while the ones with boss music (Big Bob-Omb, Bowser, Eyerok, etc.) are obviously bosses.
Super Mario Sunshine inverts the trope, oddly, as in each level you first face the local boss, and then Shadow Mario near the end.
Super Mario Galaxy and its sequel both have lots of minibosses, which are the bosses of the galaxies that precede the Grand Star stages (where the actual bosses reside).
Streets of Rage 2 and 3 had loads of these, including Jack (a knife-wielding gangbanger), Electra (the lady with the whip), the Fat Boys, and Hakuyo (the Chinese martial artist). They would often reappear in later levels either as Degraded Midbosses or in conjuction with other Mid Bosses.
Many first encounters with enemies that are stronger than the average angel in Bayonetta count as well. By the end of the game however, they become regular enemies and even some of the previous bosses become minibosses as well. The game suggests that they're different from the original bosses by giving them a different color scheme and an English name while the original versions had Latin ones.
The King of Fighters will sometimes have a character challenge you from nowhere, interrupting the normal flow. These characters usually fight alone (unlike the usual team battle), but usually have increased defense to balance it. In the console version of KOF XI, your actions up to that point determined the midboss, and if you beat them, you unlocked them.
In Wing Commander, named Kilrathi aces such as Bhurak "Starkiller" nar Caxki and Bakhtosh "Baron Redclaw" nar Kiranka in the first game, or Kur Human-Killer in the second, qualify as minibosses.
For an action-adventure game that follows the footsteps of Zelda, Okami has a miniboss cast as well. In order of appearance, they are Waka, the Satomi Canine Warriors, the Tube Foxes, Evil Rao, Oki, Nagi, True Orochi, and Nechku alone. A Bandit Spider, almost a replica of the first boss, can also be fought three times (one in each of the Devil Gate grottos that house the very difficult Multi Mook Melee matches). Lastly, judging from the tense music, the three big fish creatures that are captured at different points (Whopper, Cutlass Sword and Marlin) are minibosses as well.
There are five in No More Heroes 2: Desperate Struggle, and are fought through the optional revenge missions. They are the ones who kill Bishop (Travis's friend) after the start of the game.
In the third Fatal Fury game, Yamazaki will step in to challenge you twice, once midway and once at the end of the arcade ladder. The first battle is a one-round fight and he's not too strong but in the second bout, the gloves come off and you take him to the full length of the match. Succeed and you move onto the Jin brothers, Chonshu and (provided you do well enough against Chonshu) Chonrei.
Some of the linear Castlevania games would have this. IV had only a mid boss in level 2, while level 4 had both a mid boss and a level boss.
Castlevania Portrait Of Ruin has the Dragon Zombie that shows up in a couple of places. Unlike most other non-boss enemies, it appears the first time you visit the room and is replaced by weaker enemies in future visits.
The NES Teenage Mutant Ninja Turtles game featured Bebop as a miniboss halfway through the first major stage, who goes down fairly easily unlike Rocksteady, who you have to fight in order to rescue April.
The third NES game has a mini boss fight with Slash in the third level, while later on you twice battle a foot soldier riding a giant mouser.
MadWorld has several different ones for different areas; Big Bull Crocker, Yee Fung, Tengu, Death Blade, Big Long Driller, and the Cyber Slashers in order of appearance. They have surprisingly high health, a variety of attacks, and the ability to get into Power Struggles with Jack. Naturally, you tangle with nearly all of them in the stage leading up to the final boss (Death Blade and the Cyber Slashers, for whatever reason, weren't in on that action). In that stage, the Quirky Mini-Boss Rush turns Yee Fung into a Mook Master, has Tengu flanked by dozens of ninjas, and finally has two Bulls and a Driller.
"Devil May cry 4" had the Blitz and The alto angelo.
Anvil Of Dawn has Messengers, who function as field commanders for the Big Bad. You fight about seven throughout the game. There's also a tougher, recolored Wither Priest guarding the key to the Castellan's hall.
Resident Evil 4 has a few (with Dr. Salvador basically being a Mascot Mook) and are often fought in order to get an important item, or gain access to an area.
Dark Souls has quite a few of these, most prominently are the Black Knights. Mini-bosses in Dark Souls are much tougher than normal enemies and don't respawn, but aren't worthy of being called bosses. Some of them turn up as regular enemies later.
Traditionally, the Mortal Kombat series includes a particularly challenging sub-boss right before the final boss in arcade mode. Those who fill the role include...
Mortal Kombat: Deception changes things up a bit by having the tag team of Noob Saibot and Smoke. In the PS2 and Xbox versions, you had to unlock them, but on the Gamecube and PSP, they were available right away.
Mortal Kombat Vs DC Universe has Darkseid or Shao Kahn as your penultimate opponent. If you fight only opponents from either Mortal Kombat or DC, then you will fight their representative Mighty Glacier, but if you fight opponents from both sides, the game picks at random.
In addition to Goro and Kintaro, Shang Tsung is always your eighth opponent. The twist is that he can access some AI-only moves.