The Social Link system is both one of Persona 3's greatest concepts and one of its greatest failings. It is the all-important system that binds the social and combat aspects of the game together and gives meaning to the choices the player makes concerning the hero's normal life.The Social Link system works to do two things: give structure to the "social life" aspects of the game and to provide tangible gameplay rewards for interacting with those aspects of the game. In many ways, Persona 3 would not work if it were not for that system. At the same time, the Social Link system is far from perfect.
It may be a bit odd to do so, but I will talk about the benefits of building up Social Links before I talk about the mechanism of building them up. After all, the benefits are very important to the system; if there was no reward, then players would have no reason to even bother with the entire Social Link system and would be better off saving time by just returning to the dorm after school. Fortunately that is not the case, since the rewards for building up Social Links are very good indeed. Raising the Social Link level of a particular tarot arcana will give bonus experience points to any Persona of that arcana created through the Persona Fusion system, so any Persona created with a level 10 Link of the same arcana will immediately gain 5 levels upon creation. Because Personas are the basis of the main character's power, this means that building up Social Links provides a significant increase to the hero's power across a large stretch of the game. If you build up Social Links then you will have strong Personas, and if you have strong Personas then you will have a much easier time completing the game, thus there is a strong incentive to build up Social Links. This part of the system works very well, and all I can do is praise it.
The part of the Social Link system that I think is flawed is the other aspect: the way it forms a structure for the social aspects of the game. Namely, that structure, the method by which you build up the Social Link levels, is nowhere near as interesting and involving as it should be. Under the current system, there is exactly one group or character associated with each Social Link, and ignoring the plot-dependent Social Links, there are exactly ten significant scenes (and a slightly larger number of insignificant ones) tied to each of those groups and characters. Whenever you build up a hidden friendship value associated with each Link high enough, you can spend some time to watch one of the ten significant scenes, which raises that Social Link's level by one. This all seems good enough, but problems arise because the ten scenes that raise Social Link level are pretty much the entirety of the player's interaction with the Social Link characters.
The limited presence of Social Link characters is something that stands out in the game. Whenever you speak to a Social Link character while walking around town or school, pretty much all they ever do is give you a choice of hanging out with them or not. If hanging out with the character is not an option, then they will either say something generic (which only changes as you raise Social Link level) or simply not be present. Unlike the generic people standing around town, Social Link characters do not react to the progression of the plot. If a character who previously did react to changes in the plot becomes a Social Link character, then they stop reacting to the plot (this is very noticeable with Yukari). If you hang out with a Social Link character, but have not yet built up enough friendship to see the next major scene, then the narrator simply tells you that you spend time talking before returning home, and nothing else happens. Once you have raised a Social Link to level 10 (the state where you become lifelong friends), you no longer have the choice to hang out with that character, the character reverts to having a single set response if you talk to them that never changes, and in at least two cases the character disappears from the game entirely. In other words, Social Link characters are isolated from the plot, can only be interacted with in limited scenes that you must prompt, and they stop mattering to the game once they become important friends. I don't think I am alone in thinking that this kind of plot isolation is not very satisfying.
In addition to all of that above, another property of the Social link system is that the ten plot scenes where the Social Link character can actually be interacted with are extremely linear. At no point in these scenes can you make a choice that would change the outcome of the scene (let alone the Social Link's plot) or prevent you from building up the Social Link any further. It is not even possible to make an error that would prevent the usual raise in Social Link level at the end of the scene or cause the reversal of the tarot, no matter how much you make the Social Link character angry. There are many conversation choices available in these scenes, but all they do is control how long it will take before you can see the next scene. Regardless of what choices you make you will progress through the same story, so long as you continue to decide to hang out with a character. Again, this is something that is rather unsatisfying.
The final problem of the Social Link system is that it doesn't force obligations on the player, or at least obligations which the player has the ability to choose not to uphold. For the most part, even for things like clubs and student council, there is never any obligation to spend time with any particular Social link on any particular day, no matter what choices you make regarding social links. For the few things they ask you to do on particular days (like the summer sports competition), you are forced to go whether you would like to or not. This is really problematic regarding the "time management" aspect of the game that I wrote about around a month ago. It means that the only decisions you make are "what can I do today" rather than "what can I do today considering what I promised to do yesterday?". A good time management game (like Harvest Moon) has many aspects of the latter (such as the obligation to water crops once you have planted them), but Persona 3 has nothing of the sort. As such, the social aspect of Persona 3 doesn't really involve the kind of detailed planning and schedule juggling that can add a healthy amount of difficulty and complexity.
In the end, the only real choice you need to make in the social aspect of the game is: "who will I build up Social Links with?" Once that choice is made, the only thing you need to do is continue spending time with that character until the Social Link is at level 10. The only interesting complexities come from trying to balance spending time with several different characters at once, which is not quite enough to make up for the lack of flexibility and character interaction. The many flaws in the Social link structure are a real pity. The rewards for building up Social links are great, but even more importantly the characters of the Social Links are great characters in their own right. They are very well designed, well rounded characters with complicated personalities and engaging stories. A great deal of my criticism of the Social Link system is due to the fact that I really like the characters and wish they had a much greater presence in the story of the game itself. Characters like Kazushi, Yuko, and Mamoru should be at the center of the game's plot, not isolated away in a small corner of it.