James 5:14-17 discusses healing through the anointing of oil. Surely these days we have plenty of different types of oils but what oil was used at the time of this scripture being written?
The base was almost certainly olive oil, since that was the most readily available vegetable oil in the Mediterranean area (at least in the ancient world).
The Wikipedia article (which references various scholarly Bible commentaries) regarding anointing with oil in the Old Testament states tha for the anointing of priests, olive oil was used, mixed with myrrh, cinnamon, and an obscure herb not identified today called kaneh bosm.
Olive oil continues to be the base for all of the oils used in a Christian context nowadays: most notably the oil of chrism (also called myron in the Eastern churches), used in Baptism, Confirmation (Chrismation), and Holy Orders. A different oil, called the oil of the sick—also based on olive oil—is used for the Sacrament of Anointing of the Sick.*
Since James is almost certainly writing in the Middle East, in a context heavily influenced by Jewish culture (indeed, probably to an audience of Jewish Christians), the oil in question is almost certainly olive oil.
In the Catholic Church, if olive oil is not available, especially in an emergency, any vegetable oil may be used as the oil of the sick instead of olive oil. However, the oil distributed to parishes by the diocese will nearly always be olive oil.