Take your pet to the Water Faerie at the Healing Springs and click "Heal my Pets". You can use the Healing Springs every 30 minutes. She may not heal your neopet the first time you go, but keep going and eventually she will. Or, if you don't want to wait for the faerie to heal your pet, you can get the appropriate medicine to cure it. (Check the Hospital to see what item you will need to cure a specific illness) She will often give out free healing potions or will heal your pet at no cost; these same items can also be bought from her by clicking "See what is for sale". You can buy 1 item every half an hour.