An oil lamp is an object used to produce light continuously for a period of time using an oil-based fuel source (Wikipedia).
Filch lights his office with a hanging oil lamp. This is the only room in the castle mentioned which is lit in this way, which is possibly because as a Squib, Filch can't enchant candles to float above his desk (CS8).
Oil lamps have been used for over than ten thousand years, containing vegetable oil (e.g. olive oil in Greece and Rome) or petroleum (Persia). It is used most often when electricity is not available (Wikipedia). As electricity does not work when magic is around, it is understandable that some other kind of lamp is used instead.
However, one does wonder why wizards refer to such ancient Muggle contraptions instead of using the bright blue flames that Hermione could produce, which extended warmth and could be transported in a jam jar (PS11).
On a different note, oil lamps are still used today for religious purposes, as a source of lighting the way or to light icons or shrines (Wikipedia).