Essential oils are complex organic (carbon containing) chemical entities, which are generally made up of hundreds of organic chemical compounds in combination that are responsible for the essential oil's many characteristic properties. These characteristics may include medicinal properties, such as anti-inflammatory, healing, or antimicrobial activities, but may also be responsible for negative qualities such as photosensitivity and toxicity [1].
Here is a basic list, from Plant Therapy, covering the most common essential oil constituents and their primary therapeutic benefits.
You can find a list of the chemical constitutents on a GC/MS report for your essential oil which is available from the supplier.
For more information on the chemical make up of essential oils, consider a great reference book like “Essential Oil Safety” by Robert Tisserand or “The Complete Guide to Aromatherapy” by Salvatore Battaglia. Both of these books break down each chemical family, compounds and allow you to have a better grasp on what each does therapeutically.