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.
Knuckles from the Sonic the Hedgehog games started off as a rival to Sonic in his debut game in Sonic 3 And Knuckles, but later on, he is often billed as Sonic's "friendly rival"; in the ArchieComics, the "friendly" part was often ignored, as many of the two's conversations simply wound up dissolving into fistfights. This lasted until the "Endgame" arc, after which the two became steady friends.
Another example from the Archie comic: Geoffrey St. John, who competed with Sonic on missions and for Sally's heart. To a lesser extent (a much, much lesser extent), Antoine also originally fit the bill. Recently, Geoffrey became the Rival Turned Evil.
Knuckles gets his own rival in the form of Rouge the Bat in Sonic Adventure 2 whereas Tails battles Dr. Eggman. The former got a little better, but definitely not the latter.
Sonic, Tails, and Knuckles face their respective rivals Jet, Wave, and Storm in the Sonic Riders series, although Jet takes this further as he is the only rival in the series that can seriously get into Sonic's skin.
The Sonic Rivals series take this trope to a whole new level. The first game has four characters — Sonic, Knuckles, Shadow, and Silver — serving as rivals to each other, racing each other to catch up with Dr. Eggman. In the sequel, it's the same thing except that each rival have a companion to fight alongside with them (Sonic/Tails, Knuckles/Rouge, Shadow/Metal Sonic, Silver/Espio).
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.
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. In X and Y, this is kind of weird; the dialogue suggests that Calem/Serena chose a starter, but they never use one of the three choices during the game.
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.