Essential oils are several plant extracts which are highly volatile, hydrophobic and aromatic in nature. They are called 'essential' oil because each of them possesses a distinctive aroma or essence. Some of the examples are basil oil, eucalyptus oil, lavender oil etc.
Essential oils are used in every field including cosmetics, perfume, spa, food products etc. But the most important use of these oils can be found in aromatherapy. The antiseptic, anti-depressant, anti-fungal and anti-inflammable qualities of essential oils make them great medicine alternative to cure minor wounds, improve respiratory & circulatory systems, relax muscles and stimulate the immune system of our body.

There are more than 90 essential oils each having its own health benefits. A complete list with more information can be found out here: http://oilganic.com/essential-oil-glossary/essenti al-oil-list.htm

Yes, of course they can be harmful under certain conditions. They are very powerful concentrated compounds and just because they are 100% natural does not mean that they are always harmless. They must be diluted properly with a solvent before use (again there are different levels of dilution depending on how they are being used). There are some safety measurements and exceptions of their use, if possible please go through the book, "Essential Oil Safety" by Robert Tisserand and Tony Balacs.