The main impact of current immigration law on children is that it makes it much more likely that they will be separated from their families.
In our current system, children who are born in the United States are citizens. They are citizens regardless of the status of their parents. Therefore, any child who was born in the United States has the right to stay in the country. The problem is (from the perspective of the children and their families) that the children’s citizenship status has no impact on the status of their parents. What that means is that parents who have been caught living illegally in the United States can be deported even if they have children who are citizens.
This sort of thing could become even more widespread if the DREAM Act were to pass. This would result in a situation where children (these would be young adults) would be allowed to stay in the country if their parents had brought them to the US when they were much younger. The parents, again, could be deported.
Thus, the major impact of immigration law on children today is that it creates the potential for them to stay in the country while their parents are deported.