Anointing oil is sacred oil, blessed and charged and specially formulated for the purposes of sanctifying or elevating sacred objects or people in the service of a higher purpose.
The practice in the West seems to come from the Hebrew tradition of anointing priests and later kings and prophets with sacred, fragrant oil set aside for the purpose and forbidden for mundane or profane use. Reportedly, this oil was traditionally made from myrrh, cassia, cinnamon and olive oil with a fifth ingredient that scholars cannot agree upon, but I suspect was calamus. The oil was poured over the head or body of the person to be anointed to mark them as sacred and set apart from thenceforward.
In Christian practice, anointing oil is used much more frequently. Historically the sick were anointed with sacred oil to bring them to the attention of God and encourage healing. Today, children may be anointed at baptism with a dab of oil drawn on their forehead in the sacred oil.
In many European-based magickal practices, anointing oil is used to mark a person who has cleansed themselves in preparation for magickal ritual and/or to come into the presence of the Gods. Anointing magical tools to mark them for magical use only is also a popular ritual. Many magic-users prefer to create their own oil blends for this purpose, but Abramelin oil is popular, even without the traditions that hold it sacred, and very close to the ancient Hebrew recipe (Abramelin was Jewish after all).
In Hoodoo and Conjure, anointing oil is used specifically the anoint people while Dressing Oil is used for objects. These oils are often Condition oils, meant to help relieve or improve someone’s condition. Thus, anointing them with the oil may be part of a ritual to attract love or luck or to send away a bothersome person or bad luck. Rather than a small part of the ritual, a little dab with the finger, anointing oils in Hoodoo practice may be a central part of the magical treatment and involve a full-body anointing, or just anointing of a specific body part, depending on the condition of the person receiving the treatment. See also conjure oil