a moirail is kind of your emotional backup. so not just someone you like, but someone you rely on. you may also run across 'kismesis' -- that's also from homestuck, and in the comic it's basically foe yay, it's a hatefuck romance. but some people like me and luka use it half-jokingly to mean a friendship with trash talk and competitiveness. the great thing about these terms is that they're not just alien terminology from a sci-fi, they're useful for describing relationships we're already familiar with. so that's why they've been catching on so much.
You wanna fite? (j/k I'm just way into auspisticism in fanfic and it annoys me that its seen as so irrelevant. It has great potential for character dynamics!)
Basically it's being a mediator. In the Homestuck sense it's mediating between two people who shouldn't hatefuck. I like the Boat Trolls auspisticism Jethro and Galley have, where it's less stop two people hatefucking, it's "intervene between one person and their issues, play shrink/sounding board/support without the lovey-doveyness of moirallegiance. A moirail will give you cuddles and love and soothe you to get your emotions evened out. An auspistice will put a loving boot up your ass and tell/help you to get your shit together.
yet another homestuck alien relationship. :P clubs (also called ashen romance or auspisticeship) is when you have two people who for whatever reason need not to be in a kismesis relationship (they already are with other people and need not to cheat, or they'd kill each other instead of having a nice rivalry, or whatever), so a third person comes in and mediates to keep things from getting out of hand.
essentially 'clubs', 'ashen' and 'auspisticism' denote a three-people-romance based around one party mediating between the other two. In homestuck it usually ends up being one character mediating between a particularly nasty kismesissitude or a relationship that flips quadrants too often, in order to stabilize the other two and keep them functional through romantic drama.
Spoiler: excerpt from boat trolls rp "Evenin’, Zaps," you say cheerfully. "Pool party!" "X don’t wanna," you complain, not lowering your palmhusk. You’re about three tetrises away from beating your "using hands" high score. "Come back later." You ignore that, because it’s dumb. You perch on the edge of his pile and waggle the corner of a towel at him. “Had your brekkies yet?” "No," you growl, batting it away. "Defxne ‘party’." "You getting some weight off your bones and swishin’ around in the water some, while I do impressions of my lusus. Imma fetch you your breakfast while you get ready, then. I promised Pancho I’d help out with the vittles more." "Fuck that, X’m not hungry," you protest. Even though you kind of are, something in you refuses to give any ground. "And X hate bexng wet." "Dude, eating ain’t negotiable. I’ll make you whatever you want long as it ain’t ‘nothing’. And when we was talking about what your PT was gonna be you was all ‘gimme whatever sucks most cuz I deserve it’ so I reckon water’s perfect." You grin obnoxiously and tweak the point of his ear. "Meow!" "Yeah, well," you stall for time by slapping his hand aside. "Maybe X changed my mxnd. Maybe what X want to do now xs take a nap. Maybe leavxng me alone so X can sleep and have shxtty nxghtmares xs the rxght way to my shxverxng regret gland." "Or mebbe you’re bein’ a whiny piss baby and Imma plant my foot in your ass for wasting my time iffen you don’t knock it off. Place your order for breakfast now, you wiggler, cuz I’m countin’ down to oatmeal. Ten… nine… eight…" "No, no, not oatmeal, not oatmeal, fuckxng… okay, fxne, X’ll order, X’ll have—stop countxng—toast!” You pull your knees up under your chin so you can mutter invective into them. Whiny piss baby? You feel a biting spell coming on. You give him an obnoxiously smug and cheerful grin. “Back in ten,” you chirp, and stride out with a bounce in your step. "Wxth jam!" you call after him. "And cereal! The one wxth the marshmallow cartoon guy faces!" ^^^ jethro mediating between galley and galley's bullshit :D he also mediates between galley and pancho, though that's not so much that they were gonna hatefuck as that they just could not get along and it was driving everyone nuts.