A 'Cast' is not part of everyday vocabulary (unless you dance everyday of course), and sometimes you are given instructions to 'turn' and sometimes 'cast' and they seem very similar....
If it were really that simple of course.... you'll meet the instructions turn single mainly in the more refined English Country dances, and of course 'right hand' or 'left hand' turn. The latter are self explanatory - you do just give your right hand (or left hand) to the person you are dancing with and keeping a little bit of 'pull' in the arms, walk round each other. Your 'pull' will help them round, their pull will help you....
'Casts' are normally done on your own - so when someone says 'first couple cast down the outside', they mean that you and your partner will be separating and dancing down the outside of the set (and you'll meet again at the bottom). If the caller wants you to cast round together, then they'll say something like 'double cast' left, right or whatever.
The Elements also has some words to say about a cast - and yes, cast's are often described as sweeping to emphasise that they can, at least potentially, be done gracefully.
What can go wrong with a cast? Well, watch out if you turn too quickly, part of the idea of a cast is that you may also be getting out of someone else's way, particularly if you are going into a reel or hey. You could have people wanting to end up where you are starting from. If you do a quick turn rather than a cast and you'll end up face to face with them, both looking confused....
If you want a dance with casts going in all directions, have a look at the Guidman of Ballangigh