Anything which would have gone to a brother or sister (if they had survived the person who died intestate) now goes to their children.
For example, let's suppose that Fred (who has just died) left no widow and had no children. His parents died many years ago. (I'm trying to keep this simple and avoid 'prior rights' and 'legal rights'). Fred had a sister, Mary, and a brother, Bert, both of whom died before he did. Bert and Mary theoretically each get half of Fred's estate. Bert had just one child, who now gets that half. Mary had three children, who now share Mary's half equally, so each of them gets one sixth of Fred's estate.
It's important to note that Bert's and Mary's children only inherit when their deceased parents would have a right to inherit. If Fred had died (with no surviving wife), leaving children of his own, all of his estate would go to his children and none would go to Bert and Mary's children (since Bert and Mary wouldn't have inherited anything even if they'd still been alive).
Sourced from:
Pages 201 to 205 ('Distribution on Intestacy in Scotland') of Wills and Probate (2nd edition, with subsequent amendments), published by the Consumer's Association.
Chris