Essential oils can safely be used aromatically or topically. (Internal use is not recommended. Read on.)
Aromatic use can be as simple as opening the bottle and taking a sniff, or using the oil in a diffuser for ongoing benefit. The oil molecules have quick access to the brain via the nose and they also get straight into the bloodstream via the lungs.
Topical use is any application of essential oils directly to the skin. An easy, catch-all recommendation is to apply the oils to the bottoms of the feet. This is where your largest pores are and there are a lot of nerve endings, so the oils will quickly be absorbed into your bloodstream and carried to where they need to go. Essential oils can also be applied to specific sites, such as behind the ears, at the base of the skull, on the chest or back, or directly on a wound. It is generally recommended that the oils be properly diluted in a carrier oil before applying to the skin.
People can respond to different oils and different applications in different ways. Lavender can be very useful in promoting more restful sleep, but some people find it more effective to have it diffused in the room all night long and others prefer to rub it into the soles of their feet before bed. Sometimes a double-whammy is the way to go.
Conflicting information about internal use (swallowing essential oils in liquid or capsule form) can be very confusing. Basically, there are two schools of thought. In Europe, aromatherapy is largely practiced by medical doctors. They have the expertise to properly prescribe and monitor internal use of essential oils. In North America, very few aromatherapists are medical doctors and very few medical doctors are aromatherapists. Therefore, certified aromatherapists here approach internal use much more conservatively and generally advise against it altogether. Health Canada and the Canadian Federation of Aromatherapists do not recommend internal use of essential oils.
By and large, essential oils are safe, effective and easy to use. However, there are some safety precautions to be aware of. Some essential oils (specifically any in the citrus family) cause photosensitivity. Some oils can irritate the skin. Some are too strong for babies and young children. Some should be avoided by pregnant or nursing women. And there are some possible contraindications with medicines. Let a certified aromatherapist get you started, and feel free to ask a lot of questions.