Scottish Country Dance figures

Cast:

"Cast" means to cast down. We also say "cast up" to cast up. "Cast off" also means to cast down (also true in ECD).
We also sometimes do long casts, where for example the first couple will each just cast down on the outside for four bars, and then back up to their original place.

Lead:

Lead up/down means to take right hand in right, as opposed to the "nearer hands" we do in ECD. One advantage of leading with right hand in right is that you can turn to face the other way without dropping or switching hands.

Turn / Pass:

Turning someone means by the right hand unless otherwise specified. Similarly, passing someone means by the right shoulder unless otherwise specified (this is the same as the way you would pass them if taking right hands). They might say "pass by the right", and they mean right shoulder.

Rights and lefts:

"Rights and lefts" means four changes of rights and lefts. "Half rights and lefts" means two changes of rights and lefts.

Move up:

Just as in ECD, in many dances the first couple does something like casting off and the second couple moves up. When you "move up" in SCD, you step sideways on the sidelines; the couple moving up does not meet in the middle nor take hands with each other. If you are moving up just when the first couple is casting, or when they are finishing crossing, you can step in slightly to step around them, but you do so while continuing to face your partner across the dance.

Figure eight:

You know figure eights. A full figure eight involves passing through twice, so that you end up where you started. More common (just as in ECD) is a half figure eight, where you pass through once and end up where your partner started.

Circle round:

We circle using the slip step, facing the person opposite the circle, except in "strathspeys", which are slower dances, when we use the strathspey travelling step (which is a lot like the "normal" travelling step). In general you don't have to worry about the footwork at first, except that for circling it really will work better if you do the slip step where applicable. Remember to slow down at the end; you don't really fully do the last slip step. And, we take hands to circle round, just like in ECD.
Circling is often called "[number] hands around", e.g. six people circling would be "six hands around".

Hands across:

"Hands across" is a star (as it is in ECD). It's right hands if not specified. You take the hand of the person diagonally opposite you.

Reel:

You know reels as "heys". Not to be confused with the musical form that's a "reel" — if they say a dance is a reel, it has nothing to do with heys, but if while teaching the dance they say to do a reel, it's a hey.
We do reels more than in ECD, and sometimes in complex ways. You'll also see a "reel of four", but that's actually easier than a reel of three. (One thing to know about reels of four: after you pass the last person in the line doing the reel, and have to curve around to face back towards the line, you take a bar just to do that curving around at the end. That sentence is more likely to make sense after you actually do one.)

Corners:

After the progression, i.e. the first and second couples have changed places so that the order is 2 1 3 4, the dancing (first) couple's "first corner" is the person across the set to your right (e.g. 1W's first corner is 2M), and your "second corner" is the person across the set to your left (e.g. 1W's second corner is 3M).

A general note: You will find people taking hands much more than in ECD. "Cross" means to cross giving a hand (right hand if not specified). When people set on the sidelines, if more than one person on the same side is involved you take hands.

And as with ECD, but perhaps emphasized more, it's a social thing: when you set, you set to someone, and should look at them; when you circle, you look at the person directly across the circle; and so on.

There are also rules about which way up you hold hands, and about all sorts of things — you might think that people are being overly picky about some of these rules, but it's just the way we do it in SCD, and they're still happy to dance with you if you do it wrong. If they tell you one of these rules it's not because they're annoyed, it's because they're happy to dance with you and want to help you do it better. Don't let it overwhelm you; it's all good and everyone's friendly and trying to be helpful, not critical.

On a similar but (in my opinion) less excusable note, people in SCD are more rigid about gender roles than some of us in ECD. People whom they think are men are required to dance in the left file, and people whom they think are women are supposed to prefer to dance in the right file, and to dance in the left file only as needed (since there are always more women than men).

For reading the dances for Dancing in the Park, you might find the "instructions" more useful than the "briefs". The briefs will be available on paper at the dance; if you want the instructions you need to print them yourself. Most experienced dancers prefer the briefs, finding the instructions to be verbose. Myself I'm on the fence here.

Should you be inclined to read the instructions in advance, you may want to do so in conjunction with watching a video. http://www.scottish-country-dancing-dictionary.com/scottish-dancing-youtube-videos.html has links to youtube videos for about half of the several thousand SCD dances which exist.

I've also written a similar document for Scottish Country Dancers starting to dance English, as well as a bunch of other stuff about me and dance.