Children under the age of 5 can travel free of charge and don't need a ticket, but please remember that this means they don't have a seat reservation and so will be expected to sit on your knee if the train is busy. If you want them to have a seat reservation then you will need to buy a child’s ticket.
For children aged between 5 and 15 there are discounted fares available, these are displayed during the booking process.
One of the best value ways to buy tickets for kids is with a Family & Friends Railcard. This gives a 33% discount on adult fares and a 60% discount on child fares when you all travel together in Standard Class. You can find out more about the Family & Friends Railcard here.