Mage
A mage is a person who uses magic, either by casting spells (in which case he may also be called a spellcaster) or by paracarminical methods. The word is not usually used, however, to refer to a magical creature that only uses innate powers normal for its species—though it may be used to refer to creatures which, due to an enchantment, a mutation, or an accident of birth, possess innate powers not normal for their species. It may also be occasionally applied on some worlds to creatures of a species that does possess innate powers, but closely resembles another species that does not, though for such a creature celemologists prefer the word mock.
Terminology
"Mage" is a generic term used by celemologists; within a given world different terms may be used, and there may be separate words to distinguish mages who deal with different arcana or subarcana, or who specialize in other manners; or mages who use magic in different ways, or whose abilities come from different sources. Among the common English words for mages, or types of mages, are "wizard", "sorcerer", and "witch"; slightly less common terms include "warlock", "conjurer", and "thaumaturge". Depending on the world and the context, each of these terms may be synonymous with "mage", or have a particular more specialized meaning—or "mage" may have a more specialized meaning and a different term, if any, may be used as a general term for anyone who utilizes magic.
Becoming a mage
How one becomes a mage depends on the type of magic one is going to wield. If an individual has innate powers, he may already be a mage from the time of his birth (or from some later point, such as puberty, when his powers kick in), though it's possible he may still need to learn how to tap into the powers within him, or to use them safely or efficiently. A mage who uses spells or other types of learned magics, however, may need a teacher. This isn't necessarily the case; it could be that magic can be learned (albeit perhaps with more difficulty) through books, or by practice and trial and error. But, especially if magic is complex and hard to master (as is usually the case), a teacher of some sort may be essential, or even if not essential can certainly speed things along.
It could be, especially if mages are rare, that the usual way for mages to learn their discipline is to apprentice to a more experienced mage, in much the same way that many other crafts were learned in earlier days. (And, as with other crafts, it could be that the masters treat their apprentices as all-purpose servants... though of course this isn't necessarily the case.) A progression from apprentice mage to journeyman to master may be defined, though not necessarily by those terms, and perhaps with even more intermediate stages.
If mages are more common, they may learn their magic at formal schools. (This is possible even if mages are rare, but then the schools themselves are likely to be quite rare, perhaps with only one in the entire world.) These may be boarding schools, with the students living on campus, or they may only be places the students go for occasional classes. Or perhaps there is no official school with a fixed campus, but would-be mages learn from hired tutors.
There are many other possibilities. Magic may be something generally passed down between families—where exceptions may be possible, but for the most part sons (and/or daughters) learn magic at the foot of their fathers (and/or mothers). Magic may be taught by otherplanar beings, by fairies or demons or emissaries of gods, summoned through rituals or talismans or by elder mages who bind them to the instruction of the neophytes.
In any case, as soon as one has learned any magic, one is, technically, a mage. However, there may be other criteria to be considered an official mage by the governing authorities, or by existing mages. If there are strict channels normally followed for the learning of magic, then those who find other ways to develop their abilities may be considered renegades of a sort, and contemned, or depending on the society even outlawed. This could include, for instance, those who learn magic from unofficial private teachers when mages are expected to go to schools to learn magic, or those who hone their abilities on their own when mages are expected to rise through a system of apprenticeship. This does not necessarily mean that these renegade mages are any less powerful; it could even be in some cases that, by not following the dogma of the prevailing teachings, they discover novel magics that make them potentially more powerful than the orthodox mages.
Relations
Just like people of any other avocation, mages may or may not get along. It may be that mages generally coöperate, working together to further the studies of magic (though even then there are likely to be disagreements and rivalries); it could be that mages are usually bitter enemies, seeing each other as competing adversaries in the quest for magical power. More likely, general relations are somewhere in between, or more variable; alliances and antagonisms both exist among different groups of mages.
If multiple types of mage exist—that is, those who learned magic in different ways, or use different types of magic—, there may be conflict between them. Those with innate powers may see those who had to study magic to cast spells as untrustworthy upstarts. Those who specialize in one arcanum, or subarcanum, may consider themselves superior to those who specialize in another. Those who cast spells may think those who use paracarminical magics lack discipline and focus, and perhaps even see them as potentially dangerous dilettantes. These are only examples, however, and need not be the case. Indeed, it's possible (albeit rarer) for just the opposite to occur; perhaps one type of mage may actively seek out companionship with another type to balance their own abilities.
Organizations
On many worlds, mages form organizations to support each other and further their power. These may be called schools, or covens, or cartels, or by virtually any other such term, and may comprise only a handful of members, or may include thousands of mages from all over the world (or perhaps from many worlds). The goals of the organization may likewise vary; it may be nothing more than a social outlet, or may serve a purpose as the mages help each other in their studies and progress in magical power more quickly than they could separately; or it may go farther and seek political power or other ambitious ends. Regardless, just because mages are members of the same organization need not mean they get along; within a given mages' organization there's likely to be politics and backstabbing just as in any other such group.
Legal status
Local governments may have different attitudes toward mages. Some may allow any mages free passage and place no restrictions on their magic as long as they don't break any other laws; some (particularly those led by mages) may go still farther and grant mages special privileges and exemptions. Other governments, however, may outlaw the practice of magic, and imprison, kill, or otherwise deal harshly with mages in their interiors—or at least not allow mages to claim citizenship there, though they may have no objections to mages passing through, and may even be hypocritically eager to hire the services of foreign mages as long as they don't intend to be part of their communities. Even those governments most draconian in their policies toward mages, however, may make exceptions for individual mages who have performed them a particular service, or are somehow related to their powerful allies. In fact, it could be that the government is itself run by mages who are trying to stifle their competition.
Again, the legal status of mages may vary by the type of mage. It could be that mages who learned their craft at approved schools are legal, while other mages are forbidden from practicing, or that spellcasters are allowed but certain paracarminical magics are outlawed. Perhaps certain subarcana are permitted and others prohibited. (Illicit magics are often called black magic, though that term sometimes has other meanings.) Likewise, the government may sanction only mages who belong to certain organizations—or, conversely, may ban certain organizations with especially unsavory reputations.
Livelihood
Mages don't necessarily have to make a living through magic; many might pursue their magical studies in addition to an ordinary day job. It's common, however, for mages to devote themselves entirely to their magical endeavors. In this case, of course, they need some manner of subsistence. Some powerful mages may meet all their needs through magic, using it to create food, shelter, clothing, and whatever else they might require. Most mages, however, don't have the power for this to be an option—or simply don't like the inconvenience of it—and need some source of actual money.
Some governments may subsidize mages—perhaps on condition that they participate in the nation's defense if necessary, or that they share any new discoveries with government mages, or perhaps not asking anything in return for the benevolent purpose of supporting the magical arts. Again, however, this is not generally the case, and most mages must find a way of supporting themselves. Fortunately, where mages are not too common, this is rarely a problem, since there is likely to be ample market for a mage's services. Mages may charge to cast spells, or sell talismans... or they may engage in blackmail or kidnapping, or sending summoned or magically controlled minions to pillage nearby communities, or other more sinister schemes to get their incomes.