All I would like to add to what has been said is to advise the person trying to spin NOT to look down at his/her feet, or bend his/her body, as that will usually result in falling over.
There's a good list of spinning tips (and also tips for guys leading spins etc) on the wonderful jiveaholic site (no apologies for plugging it again) at
http://www.jiveoholic.org.uk/
Jiveaholic also refers out to another site at
http://www.eijkhout.net/lead_follow which I haven't explored yet, but seems to cover an enormous range of dance topics.
In my experience, the average lady dancer can spin better than the average guy dancer - possibly they have to, as a lot of the male spinning is optional, so maybe some of the ladies reading this can say how they learned to do it so well?