With any specialized skill comes specialized language. As a grant writer (in my other life), I work to avoid jargon that is meaningless to outsiders.
Skating parents are the ultimate inside-outsiders. Definitely an important part of the equation, but not really skaters themselves, so what do they really need to know about skating terminology and jargon? What about skaters? Sometimes in class, you'll use the colorful or description word or phrase instead of the correct one. So what is skating jargon and when is it appropriate for whom to use it?
First of all, if you're deep into it--your skater is competing in qualifying competitions-- you should at least have a passing understanding of the terms. You don't need to know how to do it, or even what it looks like, but you should know that a Salchow is a jump and a Layback is a spin.
If you're a skater at any freestyle level, even just in classes, you should know every skating term through your level; if you're a serious skater, whatever that means to you, you should know every skating term through your level and beyond. Know what a bracket is. Know what a qualifying competition is. Know the difference between ISI and USFS.
If you don't know the terms, don't pretend you do. Sadly, we got a lot of mileage recently making fun of some poor parent who posted a lesson request, written very pretentiously, about how her "serious competitive skater" need someone who "knows how to teach whirlies" (we are still not sure what this is), fix her axle (sic), et cetera, and that only serious high level coaches would be "considered." This person tried to sound like they had a clue, which they clearly didn't. We know you're not a skater or a coach; don't pretend you are (this is just good advice no matter what the discipline). You'll just sound like an idiot and squander respect going in. Give me a parent who doesn't feel compelled to know everything I do any day. I don't try to pretend I know everything about the law that my skater's lawyer-mom knows; why should you or I expect you to know everything about figure skating.
You can educate yourself. Ask the class teacher what the class is working on. Ask if you can get a brief overview sometime so you know what you're watching. Look things up online. Youtube is full of ice skating how-to videos (of varying quality).
As I said, skaters should know the terms, but with very young skaters they can be confusing. I'll often call a serpentine pattern (alternating half-circles along a straight line) a "snake" because it's more descriptive. I have lots of silly and/or completely idiosyncratic terms that I use as well-- "criss cross applesauce," "train tracks," "hooks." that only I and my own students understand. So I always also teach kids the real term; if they go to a different rink or a different coach, that coach or rink isn't going to know my made-up terms. If the skater hasn't learned the real term, they're going to be very confused.
Terminology is short hand. I don't want to have to keep repeating "this pattern is alternating half circles along a straight line parallel to the long side of the rink." I just want to say "it's a serpentine pattern." Here's a list of the most basic skating terms. Your assignment is to look them up and find out what they mean! (With a little help from my good buddy* Nancy Kerrigan!)
Edges (inside and outside)
Turns: 3-turns, brackets, counters, rockers, mohawks, choctaws
Cross-overs and progressives
Long axis, Short axis, continuous axis
Edge jump (and can you name the 3 edge jumps?)
Toe, or Toe-assisted jump (and can you name the 5 toe-assisted jumps?)(NB- Nancy says "3", but there are 5)
Spins: Scratch, Upright, camel, sit, layback, Biellman, pair, combination, A-frame
Figures (or Patch)
Spiral, (and like the Bielmann, Sasha, Kerrigan, Catch foot, etc.)
Ina Bauer (and other glides like spread eagles, Jenkins spirals, shoot the duck, etc.)
Here's a really nice overview of terms.
Some more of the mystery-- some turns are named for the shape they make on the ice: they're called 3-turns because the tracing they make on the ice looks like a 3 and brackets because they look like a bracket }. No one knows why they're called mohawks or choctaws, however. Pretty sure they weren't any actual Indian tribes involved, but who knows.
Some jumps are named for their inventors: Ulrich Salchow, Axel Paulsen, Alois Lutz, Nate Walley. Some for the program they first appeared in: Mazurkas, for instance. The loop is named for the tracing it would make were it on the ice instead of in the air. Some jumps are named for skaters who excelled at them- the Albright (mazurka), the Mapes (flip).
Spins are named descriptively, thank god, except the camel, the name of which may or may not be a corruption of the inventors' name, which may or may not be Campbell. (You begin to see the problem with terminology.)
Here's some more Xanboni on terms and skills:
General skating terms
* I don't actually know Nancy Kerrigan. But if you know her, tell her Xan says hey.