This is a very good question that ties in to both the mechanics of different types of characters and their narrative role, and in addition to those, their role in the group. There is no clear boundary where an NPC becomes a GMPC, and hence I will be explaining the traits of a GMPC rather than try to define a boundary that doesn't exist. However, it will still give you a pretty good picture in broad strokes what a GMPC is like and how it differs from a normal NPC.

The narrative GMPC

The narrative side concerns the role of the PCs --- they almost universally are the protagonists of the story the players are playing. Protagonist means "main character", and they can be seen as the main actors and focal points of the unfolding story. A work can have many protagonists: eg. Gandalf, Aragorn and Frodo are all protagonists of The Lord of the Rings (and to a lesser degree, so are a variety of other characters!), and Woody and Buzz are the protagonists of Toy Story.

For contrast, NPCs are usually supporting characters that might have a big role in the events going on, but who are not focused on in a similar way. They might be antagonists (opponents of the protagonists) or supporting characters who reinforce the protagonists' story somehow.

A narrative GMPC is a character, played by the GM, who is a protagonist similar to the PCs, complete with comparable agency and personality to the other PCs. This is often seen as an issue since this means they also claim a share of the narrative focus --- while also serving as the chief director of the story. Care must be taken so the GMPC won't steal the show and turn the game into a novel written by the GM instead of a collaboratively told story.

The mechanical GMPC

The other side of GMPCs is more tied to particular game systems and depends on the mechanical distinction between PCs and NPCs. Some RPG systems make little or no mechanical difference between the two classes of characters: eg. Wild Cards of Savage Worlds work almost equivalently regardless of whether they're PCs or NPCs. They're built using the same rules and they follow the same rules with only minor alterations.

In the other extreme end, we have games like Apocalypse World where there are very few rules directly concerning NPCs. They cannot trigger moves themselves, only be affected by the moves triggered by PCs. In the middle of the spectrum, we have games like Dungeons and Dragons where NPCs follow the same basic mechanics as PCs but have simpler features that better suit their role in the game than a full-blown PC would.

From the mechanical standpoint, a GMPC is a character who is played by the GM but uses full or slightly altered PC mechanics. However, the term is usually not used for antagonists, as building enemies using the PC rules is fairly common in games like DnD (even if it's not always advised).

The "missing role" GMPC

The third axis of GMPC is the role in the group and ties in closely with the earlier two. Some games expect the PCs to form a group with a particular composition, or are commonly played as such by convention. This can result in a "role" needed for smooth gameplay experience being missing from the PCs created. The stereotypical example is missing a "Healer", which is necessary (or ostensibly so) for playing certain systems or scenarios. A GM can attempt to remedy the missing role without forcing anyone to switch their character by creating a new character to play the missing role instead.

Whether or not the Healer (or other "missing role") is referred to as a GMPC varies a bit. By my experience, if the character is mostly a tagger-along who has little purpose or personality outside their mechanical niche, they're more often referred to as "NPC Healer", "NPC Thief", "Henchman", "Bodyguard" or something similar. Some games (eg. Dungeon World) even have their distinct rules for this kind of followers, separate from both normal NPC and PC rules. Therefore, this ties back into the other two elements: whether the character is mechanically complex enough to be considered a "PC" and whether the character is focused on narratively enough to qualify as a protagonist.

In a nutshell

Is the character played by the GM...

• a significant enough focus in the story to be considered a protagonist?


• endowed with similar level of backstory and character development to the PCs?


• played using full or significant PC mechanics, in systems where they differ from NPC mechanics?


• mechanically fully present with the group, partaking in all or most challenges?


• built to conform to a particular role in the group that's usually reserved for a PC?

The more questions you answered "yes" to, the more soundly you are in GMPC territory.

If the GM is playing an NPC that is a protagonist of the story, that is a GMPC

It isn't really a sharply defined category, but a subjective one. But I think it depends on having two other well-defined categories which precede it, examining their customary roles, and hanging a label on way the system can go off the tracks.

In most RPG systems, we have:

1. 
   

   The Player Characters. By (circular) definition, these are the characters of the players (where we are defining the GM as not a player, at least not in the same way.) But aside from that definition, in the vast majority of game systems, these are also:

   
   
   
   
   
   • The characters present in every session (generally every scene).
   
   
   • The characters that the game is about.
     
   
   
   • The characters who aren't just the main characters, but the protagonists.
     
   
   
   
   


2. 
   

   The Non-Player Characters. By definition, these are all the other characters. Logically speaking, these are then all the characters played by the GM, since the GM isn't a player. In a standard D&D type setting, this is everything from the monsters in combat, to the inhabitants of the villages or cities, to comparatively major NPCs with personality like quest-givers, authority figures, sidekicks, adversaries, etc. But also, in the vast majority of game systems, these guys aren't:

   
   
   
   
   
   • The characters present in every session (generally every scene).
   
   
   • The characters that the game is about.
     
   
   
   • The characters who aren't just the main characters, but the protagonists.
   
   
   
   

At least, not all at the same time, and hopefully never the protagonists. Yes, you can end up with games where some sidekicks are actually present in many or most sessions. And in some sense you might also say that a game is "about" the fall of an NPC adversary... but that is stretching a point.

When a GM is playing an NPC in the same manner that a player is playing his or her character-- meaning, they are omni-present, getting equal screen time to the PCs, when the game becomes about them, when in fact they become a (or the) protagonist of the game-- then you have a GMPC. I keep bolding "protagonist" because while there are a handful of other symptoms (screen time, etc) protagonism really gets to the heart of the issue for me.

So, it is subjective. But in most cases, even though subjective, you won't have much difficulty recognizing it.

Characters in RPGs usually fall into these categories:

• Antagonistic NPCs who are directly opposed to the PCs and either cause problems for them to solve or pose a problem in themselves (for example, by attacking the party).


• Neutral NPCs who react more or less naturally to the actions of PCs and NPCs but do not act on their own.


• Friendly NPCs who serve as exposition tools or provide the PCs with the resources, information and/or motivation they require in order to solve problems created by the antagonists or the environment. But they lack the means or the motivation to solve these problems themselves, so they need the PCs do do so.


• The PCs who solve the problems of the story. They are the heroes of the story and those who should get the most spotlight.

A GMPC is an NPC who starts out as a friendly NPC, but then starts to infringe on the turf of the PCs by contributing to the resolution of the problems in the story.

The main difference between an NPC and a GMPC is how you as the GM play them. Some symptoms that you are turning your NPC into a GMPC are:

• When the party is confronted with a problem, does the character propose to take a specific course of action? Which also happens to be the solution you expected from your players?


• Does the character solve problems for the party even though nobody in the party asked them to?


• Does the character have notable mechanical progression, for example by obtaining magic items, level advancement or even gets powered up by GM fiat?


• Does the character's lifestyle and mindset fit the typical "Adventurer" description?


• Is the character's sole motivation and goal to adventure with the party and have them succeed in their quest?

If any of this is true, you might be playing a GMPC.

As stated in another answer, the line is not necessarily quite sharp, but it's usually the more clear-cut examples that are worth worrying about, not the ones that straddle the line. The term GMPC is usually employed with a negative subtext, so I'll focus on signs of GMPC-dom that are usually seen as negative:

• Uses player character creation rules in a system where NPCs usually are built with simpler rules and/or with rules that deliberately make them weaker than PCs.


• Steals the spotlight. Simply put, the GMPC manages to take up a lot of the story focus, descriptions, and other such things that make the PCs look awesome, in a way that overshadows the PCs. Note that just doing its thing is not the same as stealing spotlight: a combat henchman can just take on some enemies, and a thief friend can just help with the locked chest; but if the GM is spending 10 minutes focusing on the mechanism of the trapped chest said thief friend unlocks and on the details of how two traps hidden in the lock are disarmed in the process, that's a warning sign.

Things that may be encountered in non-GMPCs in the right context, but can make the character already skewing towards GMPC skew even stronger:

• The GMPC is significantly higher level than the PCs, making them not just deprived of story spotlight, but also tactically insignificant or outright unnecessary for the resolution of a situation.


• The GMPC is given exclusive access to something necessary to the plot, such as information, and there is no alternative way to solve the plot.


• The GMPC is always right, especially due to either additions of facts retroactively, or due to having access to information that is deliberately withheld from the PCs by the circumstances.


• The GMPC tries to direct/lead the PCs without then stepping down or delegating authority or even stepping 'offscreen' (I'm pointing out the latter nuances to differentiate GMPCs from NPC authority figures and quest givers).

It comes down to what the motivation behind the character is.

If the motivation is to have a character that is played as if the GM were just another player, that is a GMPC. Which must be distinguished from an NPC party member which can be a number of things. The NPC party member can be a character that fills a missing need, such as the cleric no one wanted to play, or even a Gandalf like mentor figure who is also part quest giver, or, one game I'm in, the boss of the cyberpunk detective agency which the players work for.

Determining if a character is a GMPC is somewhat subjective. In some cases, such as the GM using the character to engage in a power fantasy or having the character outshine the PCs, it is very obvious. In some, where the NPC party cleric seems to always suddenly discover a cache of cure serious wounds wands or always seems to avoid being grievously injured, it's not so clear. (Maybe the GM is preparing to have the cleric deus ex machina the party to prevent defeat later on, or doesn't want to have to create a new NPC to fill that empty but necessary role.)

The main difference would be their role in the adventure and the adventuring party. An NPC that gets involved in combat or solving obstacles has crossed the line into GMPC territory. It is the PCs job to do those things. Unless the players have requested aid or the NPC is doing something plot essential (eg. leading them back to the bandit camp they escaped from) they are crossing the line into GMPC territory.