There are many rivalries from the Advance Wars series between COs (Commanding Officers). Eagle and Andy (one of the missions where you fight Eagle is even named "Rivals!"), Sami and Sonja and possibly Grit and Max from the first game, Adder and Sensei and Lash and Sonja from the second game, and possibly Hawke and Candy/Kindle from the third. Days of Ruin/Dark Conflict doesn't have rivalries as such; everybody generally just hates each other, except for the NRA and IDS (the two main antagonist armies of the game) who get along nicely. Except when in a reveal the player finds out that Caulder/Stolos (the Mad Scientist and Big Bad) couldn't really care less about anyone living or dying (the NRA included), apart from when it has "scientific" purposes, and essentially leaves the NRA to crumble after its defeat.
The eponymous character of Sonic the Hedgehog tends to attract a lot of these characters for various reasons.
Originally it was Knuckles. In his debut Knuckles fought against Sonic & Tails due to manipulations by Dr. Eggman. After Eggman revealed himself as the Big Bad, Knuckles teams up with the duo to stop him. Afterward, probably due to their egos, Sonic & Knuckles clashed for one reason or another, mostly due to the latter being tricked by Eggman into doing so. Their rivalry has kind of dissipated in recent years due to other adversaries showing up and Knuckles becoming more of an ally, but still pops up now and then.
Since Sonic & Knuckles were slowly becoming friends, Shadow debuted as the new, darker rival for Sonic. Unlike Knuckles, Shadow was not being tricked into fighting the heroes and was in fact a dangerous foe in his debut. Originally didn't pay Sonic much attention outside of accusing him of copying his style, but due to Sonic's tenacity and cheating death he earned Shadow's respect and acknowledgment as a Worthy Opponent. Even after Shadow's eventual Heel-Face Turn, this dynamic is retained and the two clash quite often before teaming up against a common foe.
Debuting alongside Shadow was Rouge the Bat, who served as The Rival to Knuckles. They're both treasure hunters, but it's more of a hobby for Knuckles and he guards the all powerful Master Emerald instead, while Rouge is an outright thief who attempts to steal it. This dynamic tends to be laced with Foe Yay however.
The Babylon Rogues to Team Heroes in the Sonic Riders series. Jet to Sonic, Wave to Tails, and Storm to Knuckles respectively. Since the premise of the series is racing, this is a bit more justified; that and the Rogues are just jerks. Jet definitely takes it the farthest though, being almost obsessed with proving that he's the fastest there is, and since Sonic is known as "The Fastest Thing Alive"...
Ryu and Ken from Street Fighter have been friendly rivals to each other from the very beginning of the series. Later, Sakura and Karin came along as rivals to one another.
Later games have introduced different rivals to each of these characters - Alex for Ryu in III, and Rufus for Ken in IV.
The most famous one is between the series' two resident Ninjas, Sub-Zero and Scorpion.
Sonya Blade has a major one against Kano, who she's sought to bring to justice. She and Frost have a rivalry against each other too, but neither character has ever explained why.
Kitana has one against her Evil Twin Mileena, while Jade has one against Tanya (who betrayed the royal family).
And as for friendly rivalries, there's Liu Kang and his fellow Shaolin Kung Lao.
Wario is Mario's rival, though it's doubtful whether Mario has noticed.
Waluigi is also supposed to be Luigi's rival. It's even less likely that Luigi has noticed.
Wario's actual rival is probably Captain Syrup, a villain in some games that he's starred in, whose greed is said to be even greater than his.
In Darkstalkers, B.B. Hood doesn't intend on letting any of the creatures from Makai live, but the fighter she seems to hate the most is Lilith. In one of Hood's artworks, she and Lilith are hugging, but Hood is holding a gun to Lilith's head. In another, Lilith and Morrigan are in a house, and Lilith turns around with a scared expression, looking at B.B Hood's shadow; Hood is holding a knife.
Though often misunderstood, Squall and Ultimecia indeed have an intense rivalry between them considering Squall is what turns Ultimecia into the monster she is in the future, and Ultimecia caused Squall to become the hero he is in the past.They truly were meant to battle each other for eternity.
Sora and Riku in Kingdom Hearts. They're friendly rivals throughout the whole series, but for most of the first game, Riku became distrusting of Sora and fought against him due to succumbing to the darkness, being manipulated by Maleficent and eventually possessed by Ansem. He spends the majority of the next few games atoning for it.
Hayner's and Seifer's gangs have a heated rivalry in II. Roxas and Seifer seem to have the strongest vendetta, although how much of it is simulated by DiZ's computer is unknown. The real Seifer also seems to have something against Sora, although he shows him a little more respect.
This has been a tradition in Pokémon since the beginning. The hero character always has a rival whom he meets at various points of the game. The rival in the first generation (called "Blue", officially) was a prototype for Gary Oak, Ash's rival in the anime. The hero's relation with the rival varies depending on the game; usually it's someone he's known for a long time. The one exception is in the second generation, where it's a stranger with a dark past. Most of the time, the rival is both an adversary and an ally, depending on the situation.
In the third generation, this has an interesting twist: The rival is the playable character who the player did not choose. This is also the case with one of the rivals in Pokémon X and Y, which uses a version of Calem or Selena - minus the hat, interestingly enough - depending on who your character is.
In most games, the rivals have default names or no names at all, and the player is able to name them. There are some exceptions; in Black and White and X and Y are exceptions; the Rivals there cannot be renamed.
Traditionally, the rival's starter Pokemon is the starter that has Type Advantage over your starter. The two exceptions are Yellow (where your only choice is a Pikachu, and the rival uses an Eevee) and Black and White where Cheren gets the one with Type Advantage while Bianca gets the remaining one.
Some species of Pokémon have rivalries, which are much less friendly than any the player has. Seviper and Zangoose hate each other, as do Durant and Heatmor. This is emphasized in X and Y, where if the two rival species appear in a Horde Battle, they'll fight each other instead of your Pokémon. (This is a mixed blessing, because a mixed Horde always means one Mon that's very rare for that game and four common ones, meaning that if this phenomenon occurs, you've got four Mons that are going to mercilessly gang up on the one you likely want to catch, and you can only catch the last remaining one. You have to somehow help the odd Mon out in this case if you want to catch it.)
Mana Khemia: Alchemists of Al-Revis has Roxis to the main character Vayne. So much so, that on one occasion he even thinks of competing against Vayne in a marathon, even though they're on the same team.
Kirameki High, the High School of Tokimeki Memorial 1, has a sworn rivalry with Suega High, a neighbour High School, especially in sports domains (Suega's Soccer Club is known as one of the strongest in the National Inter High Circuit, and the opponent to reach and defeat for Kirameki's Soccer Club, in Gaiden GameNijiiro no Seishun ; it's also one of the last and strongest opponents in other sports clubs' friendly matches, in the original game) ; this doesn't prevent the two to cooperate and doing a common music concert in Tokimeki Memorial Pocket.
Kirameki High also has a sort of rivalry with Hibikino High, the High School of Tokimeki Memorial 2, but it is a much more friendlier one than the one with Suega High, as students from both High Schools interact a lot and often come to the other High School and their neighbouring districts.
Between individual characters:
Tokimeki Memorial 1 has Yumi and Saki, for the Unknown Rival variety (Yumi considers Saki her rival for the protagonist's heart as soon as Saki shares a bento with him, but Saki doesn't notice Yumi's jealousy ;
Tokimeki Memorial 2 has Mei and Homura, for the Arch-Enemy variety, as well as Junichiro and Takumi towards the protagonist, if they're in love with the same target girl as the protagonist, for the straight version of the trope.
Hanna and Linda are sworn rivals who don't get along and can go into Volleying Insults, but they hold each other in high regards, Hanna admitting Linda is a faster runner than her, and Linda going for the trouble of going to Dolphan Academy to discover her, and deep down admiring her perseverance, being herself a hard-worker.
The Asian and Johan, if the Asian goes for Sophia's heart, although the rivalry is anything but friendly, as Johan is not pleased by this.
In the House of the Dead franchise, James Taylor seems to share a friendly rivalry with Thomas Rogan, hinted by the vs CPU mode in the Typing of the Dead and James' comment in the final score screen in the House of the Dead 4.
Ulysses of Fallout: New Vegas is heavily implied to be this for The Courier, with them eventually having a final showdown in Lonesome Road. He's also a case of Unknown Rival in that the Courier has never heard of him prior to the DLC.
Kyo Kusanagi and Iori Yagami from The King of Fighters. Their rivalry has rooted since the times of their ancestor. Iori is COMPLETELY OBSESSED in beating the crap out of Kyo, and while Kyo is usually dismissive, he wouldn't miss out a good brawl with Iori.
Riki is Kunio-kun's rival. In some games, such as Renegade or Crash N The Boys Street Challenge, he serves as a boss fight or foe.
Touhou has Marisa versus Reimu, and Marisa is/was (depending on whether you consider the events in PC-98 game as having happened) even a stereotypical enemy-turned-rival. There's also the considerably more violent Mokou versus Kaguya. Recently, Reimu has gained another rival in fellow miko Sanae, whose shrine is quite popular among the people of Gensokyo.
Protagonist version, Anna Hottenmeyer from Mr. Driller.
Asura's Wrath: Yasha is Asura's main rival in the story mode. Akuma / Oni becomes an even better example in Lost Episode 2. Both Asura and Akuma are so determined to defeat each other, they turn into statues after fighting for over 500 YEARS! Then they start moving again, and fight even more!
Graves sees Twisted Fate as his archenemy. They were former con men and partners in crime many years ago, before Twisted Fate turned on Graves in exchange for great magical power. After escaping his confinement, Graves joined the League solely to get revenge on Twisted Fate.
Kassadin and Malzahar have it out for each other. Both have looked into the Void and received its power, but whereas Malzahar considers it a gift he needs to spread, Kassadin sees it as an abomination that must be destroyed. They have clashed both in the League and outside of it.
Zilean and Volibear don't seem to like each other at all. This is part of a Development Gag; the real-life Zilean, the design director for Riot games, once went on record saying there would never be an armored bear hero in the game. This sparked a forum war between those who supported armored bears, and those who didn't.
Sejuani sees Ashe as this. Both come from Frejord - a harsh tundra environment with precious few resources - and both from different tribes. Whereas Ashe is willing to play nice with both the League and Tryndamere's barbarians to bring peace to Frejord and increase her influence, Sejuani is a traditionalist who believes Might Makes Right and refuses to yield under Ashe's banner.
Viktor sees non-playable character Professor Stanwick as this, for stealing the credit of creating Blitzcrank and the technology used to revive Urgot.
Hecarim and Mordekaiser are implied to be this by some of the quotes given by the altars in the new Twisted Treeline.
The Dueling Mini-Game lets a player offer another player on the opposite team a duel. If the other player accepts, the entire server will be notified of their new-found rivalry, and the winner is the one who kills (or assists in killing) the other player more than vice versa.
In-universe, Scout and Heavy, Soldier and Demoman, and Spy and Sniper are confirmed rivalries among the classes. Redmond and Blutarch, the leaders of RED and BLU, have the most major one.
From the Ace Attorney series: Phoenix and Edgeworth, Apollo and Klavier-
In Tekken, there are Paul to Kazuya, later Kuma to Paul, Kunimitsu to Yoshimitsu, Raven to Yoshimitsu, Dragunov to Raven, Hwoarang to Jin, Lily to Asuka, King II and Murduk until they become friends, Armor King II to King II. Yoshimitsu and Bryan also qualify, but they are rather Arch-Enemies to each other. There are also Sibling Rivalries, Nina and Anna, and Lee to Kazuya.
In Robopon, there's Bisco in the first game and Nick D. in the second.
In the Classic and Alpha series, every original character had a rival (doubling as a love interest) who could also be selected as the protagonist, reversing the roles. Among the originals, there's also Rai Branstein to Ryusei Date. Alpha 2 has Sleigh Presty to Ibis Douglas (and vice-versa) and Kukuru to Sanger Zonvolt. Baran Doban serves as a rival to both Sanger Zonvolt and selectable protagonist Touma in Alpha 3, and Spectra McCreedy for either Selena Recital or Viletta. In Advance, whichever of the two pilots (Axel Almer or Lamia Loveless) you don't select serves as a rival.
Sanger Zonvolt to WodanYmir, Yuuki Jagger to Bullet Luckfield, and Kyosuke Nanbu to Axel Almer in OG2. The ATX Team and SRX Team have a friendly, professional rivalry in OGs.
2nd Original Generation introduces a pair of antagonistic Choukijin pilots as rivals for the playable RyuKoOh and it's pilots, as well as the Argent-Class Boosted Children for Arado, Seolla and Latooni. Michiru Hanaten thinks of himself as a rival to Kohta Azuma. Aracari Garcia more-or-less fills this role for Sleigh Presty and Ibis Douglas, in addition to their established rivalry with each other. Throughout the series and in Masou Kishin, Shu Shirakawa is somewhere between rival and arch-enemy to Masaki Andoh. The Series has a lot of characters.
Bloody Roarsubverts this with Long. While he's said to be Yugo's rival, he doesn't care about competing with Yugo and would actually prefer not to (they're not exactly on good terms with each other). As far as Long's concerned, the acknowledgement is enough.