Like humans, fairies have their own racial and factional issues, and like humans it can be important to keep these divisions in mind when interacting with them. First thing to note is that the word 'fairy' can refer both to the whole class of being to which we belong and to our particular racial group. In general many beings may possibly identify as a type of fairy, such as gnomes, brownies, leprechauns, even pixies (much to many fairies' annoyance). I am going to use fairy here to refer to my particular racial group. Alternatively I could use the word 'fae', but that's usually reserved for a particular class in my racial group.
Speaking of pixies, one big thing to avoid is to make sure to never refer to a fairy as a pixie. Fairies and pixies are two related but different racial groups, and they don't always get along. A good analogy can be found in humans as the difference between the English and the Irish. This is probably the number one most important rule. It's often the most surefire way to really get yourself in some very hot water, even with fairies that are otherwise very friendly with pixies.
Therefor it's important to be able to tell the difference, but this can be tricky because even if they don't always get along, there is enough interbreeding and some cultural crossover to make it hard to tell. In general, pixies tend to be smaller, look more childish, often don't have wings, and often have pointed ears, while fairies tend to be bigger, look more adult, usually have wings, and often have rounded ears. But you can't always tell the difference just by looking. Especially since a good number of both are shapeshifters.
Culturally, we both can probably trace our roots mainly to elves and aos sÃ. Fairies tend to be closer to elves and pixies tend to be closer to aos sÃ. While fairies make homes in trees and such, pixies are more at home under or near the ground. Pixies often practice a slightly different kind of magic which usually is more earthy while fairy magic is more airy, though humans without magic sense might not be able to tell. Maybe these races intermingled to form the predecessors of fairies and pixies, and maybe not. Since we are practically immortal and don't have children in the normal way, it's not like we can really trace our genealogy. Plus any fairy still old to remember isn't likely going to tell anyone.
As for being called something else, not many people are going to mistake a fairy for a gnome, brownie, or leprechaun. Most likely reaction you will get if you call a fairy one of them is to be thought of as an uninformed idiot. Of course being thought of as an uninformed idiot often means some fairies will feel they can toy with you however they want, but every smart fairy also knows that seemingly uninformed idiots are people you have to keep a very very close eye on. Also, a fairy who is called an elf, aos sÃ, angel, or demon will probably think it's amusing and try and trick you into thinking they are one, maybe even shapeshifting to match.
Beyond that, fairies are divided into different factions. First of all there is the division between solitary fairies and trooping fairies. Solitary fairies (like me) tend to be very anarchistic and live in small groups or on our own. We are totally awesome and have fun all day, and are so much better then the stuck up trooping fairies. Okay maybe I am a biiiit bias. Trooping fairies on the other hand really dig their games of politics and prestige and form grand stuffy governmental courts that build and live together in cities, hold government, and make war on one another as part of their silly games.
Of course being immortal, they never really are at risk of dying like humans do (morals that join often become fairies themselves if they die), and any of them can just decide to leave and become a solitary fairy if they get board of it. There are some advantages to joining the courts though, namely they often provide a bit more training and education, they often have a much more developed economy and so have a variety of interesting goods and services for sale, plus many just find the game thrilling. Both courts are ruled by a king and/or queen, though who the rulers are frequently change as well as the exact political system depending on the whims of the rulers or their supporters.
Trooping fairies are the ones, particularly the high ranking ones, that usually are referred to as 'fae' rather then 'fairy'. This is a consequence of the fact that their courts are the main reason the word 'fairy' often refers to a whole bunch of different races, because technically the courts can include almost any kind of being, occasionally they will even invite humans to join in their games. Those that join are thought of as 'honorary fairies' and sooner or later various races that commonly join the courts began thinking of themselves as such regardless of if they personally joined.
The two most prominent and successful of these courts are the Seelie and Unseelie. In general, the Seelie court play the 'good guys' and are all about being pretty and keeping up appearances. Their political game is usually about face, looking good in public, upholding honor, those sorts of things. It's not that they aren't all scheming and looking for ways to get one up on each other, they just want to make sure they don't get caught. It strikes me as a very human game to play, and I don't for the life of me understand the appeal, but to each their own!
The Unseelie on the other hand mostly play the 'bad guy' and are all about showing off power and skill. Might makes right and the ones that have the most useful skills thrive. Those that don't have the raw power to succeed better have skills that make themselves invaluable or they won't get very far in this game. They don't put most stock in appearances or pretty speeches, but they still need some kind of politics and leadership or else they focus so much on fighting each other to oppose the Seelie. As for myself, I just don't have that kind of competitive spirit. I rather laze around and enjoy myself.
Other courts come and go as well, but they never really get the attention or success of the Seelie and Unseelie and often end up collapsing form lack of interest or joining one or the other. The vast majority of fairies though are solitary fairies like me who just don't care about the game. Though we are anarchistic, that doesn't mean that we never interact with each other or don't have our disputes, but for the most part we can kind of take care of ourselves (magic can provide for a lot), don't really need to worry about dying anyway (if we are that bored of living we can close to, but that rarely happens), and have plenty of space (due to fairyland being a extra dimensional self-expanding bubble of sorts).
When disputes get big enough they can't be solved through the parties involved settling their differences on their own, they are often settled though informal civil justice systems. One example is the tattle system. Solitary fairies tend to live in small communities or singularly populated areas called 'steadings'. Because lot of them will know each other quite well (when they don't ignore each other completely), there will often be particularly well connected fairies called 'tattles'. A tattle will seek support from the people who live in nearby steadings to become part of their 'throng'. Fairies can chose what tattle they support at any time. During a dispute the tattle(s) chosen by the parties involved will attempt to provide an impartial trial, usually by selecting judges (or often acting as one themselves) and a jury from their throng(s).
Throngs also are tasked with enforcing the decisions made by their tattle and can act like police when needed. In general if there are bigger decisions to be made, all the tattles involved in the area will get together and decide the best course of action. This type of decision is usually done by consensus, or at least 75%-80% majority. In a rare emergency they may collectively decide to just go with the majority in the sake of getting things done, but that isn't really that common. Tattles also tend to organize group projects and events, such as public works projects or holiday celebrations and other festivals. These are done on a strictly volunteer basis of course, with any needed materials and/or funding (if you can call it that, most solitary fairies use stored magic power for money) being pulled strictly from donations.
If that doesn't work, there is always the option of going to a trooping fairy court, but a lot of us despise the idea, and the courts don't like to get involved in solitary fairy matters aside from the occasional attempt at recruitment. For the record, the Seelie usually settles disputes with law and legal fanangling the way most humans do, with lawyer speak and pandering nonsense so that it ends up being more about what argument sounds better then anything else. And the Unseelie? Well they just let each party find a champion that supports them, make sure they have a fair fight, and let them go at it. Simple and efficient sure, but most of the really strong champions are picky about who they choose so you better appeal to their sensibilities (yeah right), convince them you are useful, or find a way to pay them.
And there! I hope I could educate you once again, and show how much better us solitary fairies are then those silly trooping fairies!