Never apply undiluted essential oils to any part of the body.
Not the chest. Not the soles of the feet. Never. No matter how "pure" the essential oil is, any undiluted essential oil poses the risk for skin irritation and sensitization. You may not notice any ill side effects at first and think that what you're doing is safe, however, exposure to essential oils "neat" can trigger an accumulative immune response that will show up in the future. Expose yourself or your child to an essential oil enough and then one day you'll all of the sudden have an allergic or toxic reaction to the essential oil. (This is called sensitization.)
Do not add undiluted essential oils to bathwater
Essential oils aren't soluble in water, so when you add essential oils to bath water, you're potentially applying them at full strength to your child's skin, which can lead to reactions and sensitization.
Don't add to drinks
Don't add essential oils to juice, water, or other drinks. Essential oils won't disperse in a drink because they're not soluble in water, so when you add them to juice, you're basically ingesting them at full strength.
Keep out of reach of children.
Some common essential oils are potentially toxic to small children if ingested at full strength. Always keep your essential oils out of reach of children and use oils with child-proof caps if available. Common essential oils, (yes, even when they're "pure") can be toxic to a child if swallowed neat, including thyme, peppermint, birch, rosemary, tea tree, cinnamon, clove, basil, pine, oregano, eucalpytus, and some lavenders.
Some essential oil companies (or their sales reps) propose that ingesting essential oils can cure ailments. Don't give essential oils to children to take internally, even diluted. Some essential oils can have accumulative toxicities--just because you've been ingesting them for months or even years safely doesn't mean that it's okay--you may have hidden toxicities that will appear later.
If using essential oils externally for massage, always dilute in a carrier oil like coconut oil, olive oil, almond oil, etc. Here is a general guideline for dilution:
Up to 3 months: .1%
3-4 months: .25%
2-6 years: 1%
6-15 years: 1.5%
(Adapted from "Essential Oil Safety" 2nd ed, Robert Tisserand.)