Interested in this story, but think you'd be lost in all the terminology because you haven't seen the show or looked up anything about it? This guide seeks to remedy that.
A Little Snow Fairy Sugar is an anime and manga series from 2001 and 2002. The anime was formerly licensed by Geneon, but has since been acquired by Sentai Filmworks, and published in two collections of two discs each. The manga, originally serialized in Dragon Junior, has since been published in three volumes by Kadokawa Shoten. (I have not read the manga; any differences between the manga and the information presented here can be chalked up to that, and as such, are irrelevant to the crossover.)
In the world of A Little Snow Fairy Sugar, the weather is not made by complex systems involving heat, moisture, the rotation of the earth, and other factors, but is responsibility of small humanoids called season fairies. This magical weather control is achieved through the playing of special musical instruments. Each season fairy is responsible for a particular kind of weather within an area, but it is possible (and canonically happened) for the season fairy of one area to fill in for the same type in another. It seems as though it is normal for an apprentice season fairy to aspire to be the same type as a parent, but it is not unheard of (and again, canonically happened) for an apprentice to switch types, even late in training.
There are three kinds of season fairies that we see: apprentice, full-fledged, and elder. The apprentice season fairies tend to be children; canonically, the three we see most often are about 9 years old. They take up temporary residence in the human world until their flower, which they are supposed to plant as soon as possible, fully blossoms, indicating that they are fully-fledged. A major part of the process is finding "Twinkles"; this goal is frustrated by the fact that while all the full-fledged season fairies know what they are, the apprentices do not. And no one who does know is allowed to tell, as finding out for yourself is part of the process.
The normal full-fledged season fairies are already responsible for their area/weather type combination. The ones we typically see are Turmeric and Ginger.
The Elder, the leader, is a special case: he can make any kind of weather. Among his duties is personally training apprentices.
Fairies can be seen and heard by, naturally, other fairies, and can be seen and heard by other non-humans. They cannot typically be seen by humans; however, humans who will be or have been bonded to a season fairy can see any season fairy, even if it's not that one. (Saga, the main human, saw a snow fairy when she was a few years younger than she is outside of flashbacks, and until Sugar becomes full-fledged, she can also see her fellow apprentices, the full-fledged season fairies, and the elder.) Any human can feel them, as well as see things they do... such as eat the soup right off someone else's spoon. (Whether or not season fairies can be smelled or tasted is thankfully not explored.)
As you might guess, being a person who can see fairies in a world where this is rare and fairies are not typically believed in even by most 11-year-olds can be rough, particularly when one keeps provoking you to respond. But even humans for whom this is true can find themselves getting attached to the little guys, and vice versa.
So, that's the primer. If you have any questions pertaining to A Little Snow Fairy Sugar canon, particularly as it pertains to this fanfiction, ask in the comments and I'll see if the answer warrants inclusion in the primer itself. Otherwise, I'll likely just answer in the comments.
Enjoy the fic! If, that is, you haven't already and this primer has gotten you ready to.