Essential oils are the natural extracts of plants, usually flowers, herbs, or trees. They can be rubbed or massaged into your skin (topical therapy) or inhaled through your nose (aromatherapy). They are used to improve physical, mental, and spiritual health. Since so many medications need to be avoided during pregnancy and breast feeding, a natural alternative like essential oils would seem to be a good idea. But as with all things to do with pregnancy or breastfeeding, caution is advised. [1,2]
What Could Go Wrong?
Essential oils are very concentrated extracts. It takes over 200 pounds of lavender flowers to express one pound of oil. Concentrated essential oils have a wide variety of active chemicals and the concentration of those chemicals can vary between plant species and even between plants. This makes the absorption and effects of the chemicals on you and your baby unpredictable. [1,2]
The FDA does not regulate essential oils, so there is no way to know what you are actually inhaling or rubbing into your skin. Aromatherapists, who use essential oils for aromatherapy and topical therapy, are not state-licensed. In the United States, the National Association for Holistic Aromatherapy (NAHA) provides standards, certification, and guidance for use of essential oils. 
NAHA admits that essential oils may cross the placental barrier and get into breast milk and that using these oils during pregnancy and breastfeeding is controversial. NAHA and the International Federation of Professional Aromatherapists [IFPA] have published guidelines for aromatherapists on the use of essential oils during pregnancy. Both have a fairly long list of oils that should be avoided. [3,4]
According to the National Institute of Environmental Health Sciences, recent studies on two commonly used essential oils, lavender and tea tree, have found an increase in levels of estrogen-like activity. When these oils are rubbed into the skin, they may cause growth of breast tissue in young boys. There are no studies on the effects of these oils on a fetus or a breastfeeding baby, but it makes sense that exposing a baby to abnormal levels of female hormone is risky. 
Since essential oils have not been studied in breastfeeding or pregnant women, any effects they may have on a developing or breastfeeding baby is unknown. If you add in the lack of any strong evidence for benefits, you would have to say the risks probably outweigh the benefits. In other words, if an essential oil was an FDA approved drug, it would not be approved for use in pregnancy or breastfeeding. [1,2]
Although there are no studies in pregnancy or breastfeeding, there have been studies on benefits of essential oils outside pregnancy. Many of these studies have been done in people with cancer. They have shown mixed results. A few studies have found that essential oils may relieve anxiety or nausea and vomiting. A few have found some improvement in sleep or mood. Other studies have failed to confirm these benefits. According to the National Cancer Institute, there are very few side effects when essential oils are used as directed. 
NAHA and IFPA, both believe that some essential oils can be beneficial and safe during pregnancy as aromatherapy or topical therapy. They claim benefits for frequent urination, nausea, vomiting, heartburn, hemorrhoids, stretch marks, varicose veins, and water retention. 
You can find a complete list of possible benefits and the essential oils that may provide them here. You should also check the list of essential oils to be avoided. Keep in mind that the list was compiled before information on estrogen-like activity of lavender and tea tree oil was available, so you should add them to the list of oils to be avoided.
Bottom line on essential oils during pregnancy or breastfeeding is that there is a small amount of risk for a questionable amount of benefit. If you still want to try an essential oil, talk to your pregnancy care provider, or if you are breastfeeding, your pediatrician. If you get the go ahead, work with an aromatherapist who follows guidelines from NAHA. You can find a certified aromatherapist at this NAHA website.