If you are looking for something to make the carpet thicker, thus making it softer for children, an underlayment is not what you need.
An underlayment is a thin foam sheet that is typically used underneath wood flooring. What you are going to need to look for is a rug pad. If you go to your local home improvement store and ask their flooring department, they will show you the difference in the two products. The underlayment comes in long rolls and is a thin foam material; whereas carpet padding is typically sold in pre-measured square foot sections (ie. 6'x8' section) or some stores may custom cut a pad for you if you would like them to (this could cost you a little more). The carpet padding comes in lots of different sizes and lots of different thicknesses, so you can determine what is best for your family.
If you are unsure of these two options, you can research the two terms on the home depot website; it will give you more clarity on both products.
I have used carpet padding, and it makes quite a difference in the softness of the rug. I will also tell you that @lefty is absolutely correct in that you need to use some type of carpet tape or adhesive to adhere the carpet padding to the rug. Some padding has one side with a rubber grip on it, and this is the side you put on the floor to keep it in place. If it does not, use the carpet tape on the padding as well to adhere it to the floor so that it does not move around while your kids are playing on the carpet - otherwise it will happen, and they will fall; it happened to my daughter, and then we went out and bought 2 rolls of carpet tape.