One thing to clear up when discussing essential oils is that the word 'essential' does not mean indispensable, it actually means 'the essence of'. It is an extract of a plant that 'captures the essence of its fragrance' in an oil so that as the oil evaporates the aroma of the plant is carried into the immediate area. They are used in perfumes, cosmetics, soaps and other products, for flavouring food and drink, and for adding scents to incense and household cleaning products. They are often marketed as a healing product especially in aromatherapy. The oils are obtained by either distillation or by using solvents to extract the specific chemicals that we recognise as the smell of the plant.
Some are used as flavourings in food production but most people will come across them when offered for aromatherapy treatments. The practice of using aromas in medicine is very old, going back to the Ancient Greeks and possibly further. Even in medicine today there is a place for it, think of menthol as a decongestant, but alt-med aromatherapists offer extensive lists of smells and concoctions aimed at treating numerous diseases, all without any real evidence of efficacy.
Though nice smells are, well, nice, and can help subjects feel relaxed and less stressed, there is no reason to think that there is any other real health benefit to essential oils. Indeed there may be some real dangers. Many are released through burning candles, with the inherent risk of naked flames in the home. Also, some of the oils can have an irritant effect on the skin or if inhaled. Some very common oils like eucalyptus are extremely toxic when taken internally. So the takeaway is they make your home smell nice when used carefully but that's about it.