Fantasy RPG's are replete with creatures from the realm of Faerie. Some settings are more so than others. There are even games that refer to things like Centaurs, Cyclops and Ogres as Fey, though the term is mainly reserved for the wee folk. Satyrs, Dryads and Nymphs are generally human-sized, but there are a plethora of Leprechaun-sized Fey running around, and then there are the "dewdrop fairies" - Sprites, Pixies and the like - which can amount to a pretty long roll.
A perusal of various creature catalogs from different games might well leave one scratching one's head as to the differences between the various types of Fairy and Sprite. After all, many of them are small, fly and can use magic of some sort or other. While the subtle differences may be amusing to a fantasy naturalist, at the game table, they can become a liability.
To get the most out of using Fey while avoiding extra trips through the monster books, it is wisest to use a few, easily distinguishable Fey in your games. If you have played a game for 30 years and still don't really know the functional differences between a Pixie and a Sprite, then you probably don't need to use both at the table. On the other hand, if you would like to use both, then you need to have versions that you can tell apart with relative ease. "Fern Gully" and "Tinkerbell" got away with only one type, but if you want to have the sort of variety that games usually encourage, then some sorting is in order.
In "Basic D&D," Pixies are naturally invisible, attack with daggers, and can only fly for a half hour or so before needing to rest for a while. Sprites in the same rules never engage in violence but work together to cast a curse at those who would seek to harm them. Sometimes, they will use their magic just for a practical joke.
While the differences in magical powers are clear between the two Fairies, a subtler difference is one of attitude. Pixies tend to live and let live but won't hesitate to fight, while Sprites, never ones to take up arms, are nonetheless not above causing a bit of mischief. These outlooks are more important to how the creatures will be played than are their raw stats. A helpful Fairy encounter will likely involve Pixies, while a GM who wants to harass the players will have them stumble upon Sprites.
Likewise, the Brownie and Leprechaun have similarly specialized roles. "AD&D" defines the former as Lawful Good and states that they will help similarly aligned creatures, while the latter will most likely try to steal from adventurers just for a gag. Their magical powers differ, though perhaps they are too manifold to keep track of - Brownies can dimension door and perform some distracting magics, while Leprechauns are known for polymorphing non-living objects and for their own forms of prestidigitation. Their powers should probably be simplified for conversion to a simpler rules set like "Basic D&D." But, like the Pixie and Sprite, it is their attitudes, not their powers, that truly define them.
Satyrs, Nymphs and Dryads should also be distinct enough in how they are played so that, for example, a Dryad is not simply a Nymph that has to stay close to her tree. Satyrs are mischievous like the Sprite and Leprechaun, but they are more interactive with Humans and their ilk. Whereas the smaller Fey tend to harass the 'big folk" with tittering laughter, Satyrs are more inclined to seduce and befriend. Being of Human size, the goat-like charmers are better able to relate to PC races then their more mystical cousins.
Nymphs can be played in a similar fashion to the Satyrs, or they can be extremely shy and retiring. The trap many gamers fall into is making them too much like the Dryad. Remember, no matter how much a Nymph loves nature, she is not tied to one spot like the Dryad is - try to make her adventurous! The propensity that some games have to give them the power to strike onlookers blind or dead is a shame, since all that beauty and Charisma go to waste.
Nymphs might be more inclined to travel, at least to different parts of their wood, and to have a somewhat more cosmopolitan outlook, while Dryads are as much plant as they are humanoid. The latter tend to be more introspective and to look at things in terms of natural cycles. They may even be a touch fatalistic. This is not to say that the two don't share gossip and tea; only that the players should be able to tell pretty quickly whether they are talking to a typical Dryad or a typical Nymph. And while we are at it, there is no need to have Dryads be covered with bark - this cosmetic change in later games is no fun, and is a poor alternative to actually giving the two creatures different personalities.
As for the proliferation of other Fairies like Atomies, Grigs and the like, be sure that any such are used sparingly. The list above already numbers seven, four of which are of the small variety. Grigs look like crickets and play fiddles, which can always be fun, though in the end they play much the same role as the mischievous Sprites. They might be nice for a change, but don't take things too far, or you will start to lose track of you Fairies, and your players will too.
As for the larger creatures, like Centaurs, Cylcops and such, be very careful about diluting the Fey concept. The charm of Fairy is that these are strange creatures with different goals and aspirations than we have, and that we can't truly understand them. Efforts to make Fey more accessible, like in "Fourth Edition D&D," risk taking the magic right out of them. It's one thing if you are running an alternate concept setting, like "Fairy, Queen and Country," but if you are shooting for a more straightforward fantasy trope, then you don't want to put pointy ears and Fey origins and every fifth creature in the book.
In the end, keep things simple, keep them magical and keep them fairly rare - and your Fairy creatures will serve you and your game well.