The AllEars link bigorangeandy seems clear to me. Since different info is coming from different sources, here's the text:
EXCHANGING OLD CHILDREN'S TICKETS Many people find themselves in the situation of going to WDW with their young children and having unused days left on their tickets when they return home. These tickets are put away, often for many years, until another trip to Disney is planned. But what can you do with these tickets since your then child is now a teenager or even an adult and obviously can't use a child's ticket now?
What you have left will depend on how this transaction is handled. If you have a brand new, completely unused child's ticket that you bought years ago you will only be able to apply a dollar value equal to the price you paid for that ticket towards any new adult ticket that exceeds the price of the old one. This is your only option with an unused child ticket.
But if you have a partially used ticket, you may take that ticket along with your child who is now a teenager or older to a Guest Relations location at the major parks or DTD. If the Guest Relations CM is satisfied that the dates of the original ticket and the current age of your child make sense, you will have the leftover child's admissions exchanged for the identical adult admissions at no further charge to you. The "child" must be with you or you will be unable to do this.
Making sense of the dates means that if you bought and used the child ticket in 1994, then your child in 2004 must now fall in the 13-19 year old range. If you bought and used it in 1984, then the "child" must now be in the 23-29 year old range, etcetera etcetera. If they are not, then Disney reserves the right to offer you nothing more than the dollar value of the unused admissions towards a new adult ticket.
A note on this: if your child is now 11 or 12 years old, you can continue to use the old child's ticket as is and have no problem at the gate. The only time that you may run into a problem is when you have an older teen trying to use it. That is when you should exchange the child's ticket. Don't bother going to Guest Relations for an 11 or 12 year old, just use it as is.