This trope mostly crops up in Action RPGs (although a turn based RPG might use something similar and team based sports games also tend to favour it). It's when the Player Party are controlled by the game's AI, like any other NPC. This allows games which don't have turn based mechanics to still let the player control all the members of their team at once.
Most examples will allow the player to control one of the characters directly. This might be The Hero or the player might be allowed to switch between each member (being exclusively able to control a less important party member is rare). They usually allow the player to change the party's overall tactics and have the characters play defensively, offensively and other broad tactical options. Even more advanced games will make the AI fully customisable, with the player able to choose what abilities and tactics their party will use, how to react when low on health, etc (other games might give the AI the same complexity, but choose to keep these set in stone as per the party members' personalities). The player might be able to influence the behaviour of AI teammates with Squad Controls.
One of the main limitations of this mechanic is (obviously) how good the NPC AI is. In theory, the party's AI has to be slightly more advanced than the enemy AI (both to make sure it doesn't get in the player's way and to make sure that it can function in all situations, unlike enemies which only need to be able to function in areas where they appear). Fortunately for the sanity of both players and developers alike, game AI only needs to appear intelligent. Thus very basic behaviours in enemy and ally AI can still give the impression of two intelligent entities duking it out when they interact.
Another disadvantage is that the player might not feel that they're actually playing the game (indeed some games with this mechanic even have the option of allowing all the characters to be controlled by AI and let the game play itself). Of course, if the emphasis is on stat building then it's arguable that the player's playing the game for that aspect....
Compare Guest Star Party Member (who might be only controllable by the AI in a game where this isn't normally the case), All in a Row (for when the party follows the player around on their own, but not in battles). This mechanic is a close cousin of the Tactical Shooter genre, which often features AI controlled allies but generally has less focus on character customisation (as found in an RPG), favours hard realism and usually expects the player to micromanage.
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Final Fantasy XIII uses this, the player directly controls the party leader during battles and can only dictate which kind of actions the other two members take (support, tank, DPS, medic, etc). Final Fantasy XIII-2 adds the option of switching the character you're controlling directly mid-battle.
Final Fantasy XII used a similar system where the player sets up the AI for every party member (including the leader). The camera stays focused on whoever is set as the party leader (unless they die, in which case the game pauses and has the player choose a living leader), but the player can issue commands to anyone or even turn off the AI controlling them and manage the entire party's actions directly.
Final Fantasy X's Blitzball minigame used this, only the character holding the ball was under the player's direct control, everyone else on the team followed whatever formation the player had set.
Kingdom Hearts, to the degree that Donald and Goofy are barely noticeable.
Kingdom Hearts 3D has Donald and Goofy replaced by Mon party members. The right Dream Eater with the right disposition can prove itself to be far more helpful and supportive than Donald and Goofy ever were.
In Fortune Summoners the player controls up to three characters (controlling one directly and having the option of choosing the other two's tactics). In addition to switching in battle, the player can also switch between different characters to solve puzzles. There's is an unlockable option to let the AI control everyone.
Mass Effect had this in all three installments: you could decide on their armor and weapons loadout and give them orders in combat but only ever directly controlled Shepard.
Knights of the Old Republic and Dragon Age both allow customization of the party members' equipment and tactics, and allow direct control over all characters. AI scripts direct characters that are not being directly controlled, and can be customized by the player.
Neverwinter Nights allowed you to choose a henchman to follow you, and ... that's about it. You had no control over your henchman, which often led to Artificial Stupidity including randomly running off to attack enemies halfway across the map. The first expansion added party inventory control, and the second allowed you to have more than one NPC party member. In contrast Neverwinter Nights 2 followed the KOTOR model, and the second expansion even allowed you to multiclass your NPC cohorts.
In Secret of Mana, with the exception of spell casting, you can only control one character at a time (although it allows co-op play where up to three players control a character), but the offensive action of the ones you are not using can be changed with a system called the "action grid", which allows the player to switch between offensive/defensive behavior on one axis and the range at which they engage the enemy on the other.
Other entries in World of Mana uses this trope to different extents (with better or worse AI).
In Xenoblade, you can only control the leader unless it's a chain attack or after warning a party member, after one of Shulk's visions.
Star Ocean: Till the End of Time allows the player to control one character, or the entire party, by setting all characters to "manual" in the options menu. In addition, you can set the behavior of AI controlled party members from a selection of tactics, with some being unique to certain characters. This works, for the most part, but every so often....
Pretty much all of the Tales Series games do this, with varying levels of complexity depending on the age of the entry. Most of them include preset orders for defensive and aggressive behaviors, which can then be further customized in the strategy menus. You can also choose to enable and disable certain of your party members' abilities, and most entries allow you to choose how often they use special techniques, and in some cases what kind (for example, you might be able to set the healer to focus on conserving mana, healing everyone, or casting a lot of support spells). You can also set their default distance from the enemy when they enter battle, how closely they choose to engage the enemy when actually in combat, and sometimes what kind of enemies they focus on attacking (same as the player, different from the player, flying enemies, etc).
Tales of the Abyss allows the player to switch the controlled character in the middle of battle after gaining a particular item.
Tales of Vesperia is similar to the Star Ocean example, in that it also allows you to set the behavior of AI controlled party members. However, it gives the player far more options to work with, from selecting formations, setting the distance AI team mates should maintain between the Player Character and the enemy, and whether to allow them to use items (and how often).
Of course, at the same time a second, third, and fourth player can control the respective party members, usually with better intelligence.
Tales of Phantasia uses a simple method of AI customisation where you can switch the abilities you want them to use on and off.
Infinity Engine games all give the player the option of letting their party be controlled by AI (although micromanaging them is a better option during boss fights);
In Evolva, you can only directly control one of your four Genohunters at once. You can switch the active Genohunter at any time. While controlled by the AI, the other Genohunters tend to follow the active one and attack any nearby enemies, although you can tell them to go to a place, attack a specific enemy, or stay where they are.
You only directly control your squad leader in The Bureau: XCOM Declassified, while your allies are mostly AI controlled. You can give them orders as to where to stand and which abilities to use, but for the most part they fend for themselves.
First Person Shooter
In Half-Life: Opposing Force, you're usually teamed up with a squad of AI soldiers which you can direct, and use to perform certain functions (but you can't directly play as one of them). Also in Half-Life 2 you take command of a squad of AI resistance fighters.
Clive Barker's Jericho started with squad leader Captain Ross as the one controllable character in a squad of seven. After he gained the ability to Body Surf, you could swap control over any squad member available on a given map.
In Sonny, you always control only the eponymous character, with your other members being AI-controlled. In the first game, there's three settings: Defensive, Tactical, and Aggressive, which can only be changed in between battles. In the second game there is five settings: Phalanx, Defensive, Tactical, Aggressive, and Relentless, which can be changed at the start of every turn.
In the Ace Combat installments that give you permanent wingmen, you can usually select their planes and give them orders but you only steer your own plane on missions. Ace Combat 5: The Unsung War is the only one, however, where you can speak of a "party" (it had three wingmen as opposed to one in other games).
In NBA Jam, you control the player who has the ball and the rest are AI driven (successfully passing would shift control to the player who caught it). You can freely switch between players when the opposing team has the ball. The game had a multiplayer mode which allowed up to four human players to control the team in the arcade version (leaving only one AI), however.
In Midway Games' Pigskin, player's joysticks only control the team captains, and their teammates follow A.I. strategies based on Squad Controls. The punch button, however, makes every team member throw a punch.
In Ogre Battle you could control your leader, but not the rest of the group directly.
The recruitable party members in Fallout and Fallout 2 allow you to customise how close or far they stay from you, how often to use drugs to heal themselves, and how to use their weapons.
In Persona 3 you can choose your party's style of attacks (offensive, healing, general) but not control their actions directly. It's also the default for Persona 4 but you can switch to manual control. The ability to switch to manual control was also added to Persona 3 Portable.
The NES version of Dragon Quest IV uses this as of Chapter 5, with the hero the only directly controllable character. This leads to quite a large amount of Artificial Stupidity, especially when Kiryl keeps spamming low-accuracy one-hit kills instead of healing. The option for direct control is included in the DS remake.
Used in the Pokémon Mystery Dungeon series where until you get the credits, you are restricted to only directly controlling the first pokemon you start with (the human-turned-pokemon). You can manipulate the AI of the rest of the party quite easily but you never directly control the rest of the party, only the leader.