Losing a pet can be a difficult thing for adults, as well as children. I hear many people brush off the importance of a loss of a pet to a child, saying "kids are resilient". However, when I talk to many adults, they can vividly remember the loss of a pet. Worse yet, they may be still angry at the lies that their parents told them years and years ago. One of the biggest non-truths that parents tell their children is that "Fifi went to a farm". Eventually, all children grow up, and find out that was a lie. Rather than dealing with the loss in the "here and now", they feel betrayed by their own parents.
So, I would start off by saying to be honest with your child. Tell them that the pet has indeed passed away.
In dealing with death with children, I have found that it is important to let children know that pets die when pets are :
1) Really, really sick
2) Really, really hurt
3) Really, really old
For me, it is important to tell them that the pet is "really, really" sick, etc; so that they don't get nervous every time that another pet is sick; thinking that the pet may die.
Let them go through all the stages of grieving.
Be empathetic, and listen to them. You don't have to "solve" their healing. Just listen to them talk, without trying to "make it all better". It will get better, if they go through the stages of grieving.
"Mirror" their feelings, by first listening to them; then recapping what they just said. This way, they were heard and understood.
Let them cry, for as long as they need to cry. Don't stuff the feelings, by trying to coax them to stop crying. Again, you don't need to solve their pain. You just need to be their for them.
It may be a good idea to have a memorial service to celebrate the life of the pet. You can gather pictures of the pet, and talk about the things that you loved about that pet. Ask the children if they want to draw pictures of the pet. Have a candlelight ceremony in honor of the pet. RainbowBridge.com has a Monday night virtual lighting ceremony, and pet loss support.
If you need to talk to someone more about pet loss, I would recommend Susan Dowd Stone, a licensed, clinical social worker and an animal companion bereavement specialist and advocate. You can view her website by going to Pet Loss Help