Many fantasy games postulate a society where effective magic is an established part of the world. Rather than being limited to lonely towers, magicians are in positions of power and authority: they are forces in warfare, advisors to kings, and have powerful guilds. Given this, a common question for fantasy worlds is "How would real, effective magic change society?"
In general, I think it is important to point out that different magic systems will likely have drastically different effects. There simply is no answer to the general question of what "magic" does. It depends very strongly on the specifics of how magic functions and the exact spell list.
In general, RPG magic systems tend to favor small bands of elite professionals over large armies and institutions. Of course, this is unsurprising given who they are written for, but there are many logical consequences to this. While there are rare exceptions, spells in RPGs are much easier to apply to sneak in and attack a fortress rather than defend it.
Besides the tactical implications for fortress design, this has a general social effect. The level of organization of a society (i.e. bands, tribes, nations, empires) is strongly controlled by how defenses work. As a parallel, consider the human body. The human body has skin and specialized immune system cells which defend against pathogens. This allows other cells to specialize in other functions. If bacteria could easily bypass the defenses, then the body (i.e. multicellular life in general) simply would not function. All cells would have to work to defend themselves, which limits their specialization.
By the same token, allowing bands of teleporting or invisible combatants makes large empires much less stable. If the empire tries to lean on a small tribe, that tribe can directly attack the emperor. Neither the leader nor the citizens can rely on an intervening army or walls to assure stable life. Indeed, this is often the aim of fantasy RPGs: allowing a few heroes to break into the palace and defeat the emperor in the name of the underdog. However, the conditions which make that possible make it unlikely that such an empire would ever arise in the first place.
Making the spells rarer does not negate this problem. Indeed, in some ways it makes the problem worse. If invisibility is very rare, then an invisible person is much more powerful -- because countermeasures are unlikely to be developed or maintained. Even if it is known about in theory, guards are unlikely to consider it in their day-to-day actions. Thus, in many ways making invisibility rare increases its effect. Whoever can muster up an invisible attack force has an absolute deterrent of a sort.
These spells open up information in a highly unhistorical way. The development of long-range communication in history depended critically on broad resources and organization: i.e. mail systems, chains of signal towers, and telegraph lines. RPG communication magic generally requires no equivalent: just magic at one or possibly both ends. Note that communication spells don't need to be pure communication. For example, long-range teleportation is communication which takes only one side magic. Clairvoyance can be communication if both sides have it or if one can act somehow on the target. High-speed, reliable messengers like birds or demons may also work.
By the same token, many spells can be used for information gathering. Invisibility allows local information gathering, as does shape-changing into a non-descript form like a bird which can survey things. This also is highly unhistorical, in that it allows for extremely reliable intelligence. While there is illusion magic in RPGs, the cost of maintaining an illusion over a long time (even a day) is almost always prohibitive. Thus, the illusion usually will not stand up to extended surveillance. It can be used for tricks on the battlefield or in a meeting, but not to alter the strategy over the days and weeks leading up to that interaction.
Tactically, both of these change large-scale battles enormously. Historically there were many battles which were won or lost based on a key piece of information. With reliable intelligence, there would be little chance of an unexpected result where the lesser army prevails. War becomes more of a sure thing, and thus most likely rarer. In general, sides engage in battles because each thinks itself capable of winning. With less chance involved, surrender and negotiation are more likely.
Culturally, it seems these will tend to open up communication and trade among distant people. This suggests a balkanized group of related tribes and clans are more likely, rather than large empires and monolithic cultures. Scrying I suspect would tend to eliminate dissent and heresy within a culture, but that depends on many factors of how it is used and controlled.
In general, I think that the details of these are less important than simply the fact that a few key individuals can have a lot of military power. It seems to me that this is liable to produce an effect similar to armored cavalry in medieval Europe. With more of the battlefield determined by a few individuals, you tend to have more set-piece battles between the principles rather than massed armies.
The nature of spells will have some effect. For example, fireball in D&D is an extremely potent area spell. Mechanically this would argue for dispersed troops like what happened with modern cavalry. However, a dispersed mass of levied soldiers is essentially useless. It seems to me that such spells will tend toward fights more like duels or the clashes of armored knights. The leading heroes of either side agree to fight it out, and whoever is defeated agrees to accept the terms of losing.
I have tried at times to speak in generalities, but really that is impossible. Most of the social effects of magic will depend critically on the particular system used and particular spell descriptions. For example, when I looked at D&D (3rd edition), it seemed to me that most of the social effects would spring from just a few key spells. "Plant Growth" is a 3rd level spell with enormous area and duration (1/2 mile radius and one year), raising crop productivity by 33%. This has a very major effect. "Resurrection" simply by the nature of its existence would vastly change attitudes toward death and religion. However, most spells would simply have no particular effect (i.e. Charm Monster, Blink, etc.).