A well-drafted will should specifically state what should happen if a beneficiary dies before the testator does. So, for example, the will of a parent (in a situation like the one you've described to us) should includes something along the lines of this:
"I give one half on my estate to my son, Arthur James Peartree. Should the said Arthur James Peartee pre-decease me, I instead give one half of my estate to my daughter-in-law Mary Helen Peartree".
Where such a term is present in a will, then it will apply if the son dies before the testator. However if whoever drafted the will failed to include such a provision, the gift that was intended for the deceased son will be shared equally between his children, if any.
If he did not have any children, and the will was drafted in such a way that there's a 'residuary beneficiary' (i.e. someone who receives whatever's left over after all the main provisions of the will have been carried out), the residuary beneficiary will receive the gift instead.
If there's no residuary beneficiary, then a 'partial intestacy' is created, with the relevant part of the estate being dealt with as if the testator had never written a will at all (following the rules from New Judge's link above).