What is the difference between natural oil finish and a stain on wood furniture?

I am having my dining room chairs refinished and my refinisher gave me the option of natural oil finish or a stain. He somewhat explained the difference, but I'm not sure which I should do. My only concern is that the chair be refinished to the exact same color/stain/finish they are now. I do know there are two different types of wood on the chairs. The sides and back are walnut and the rest of the frame is pecan. Any advice?