The short answer is no, those express two different things.
PCI is a very old standard, hardly ever used anymore for graphics cards. But your motherboard has 2 of those slots. PCI Express (PCIe) is the modern standard for expansion cards, and it has so far undergone three revisions. First was PCIe. Then PCIe 2.0. Then PCIe 2.1. Now there is the new PCIe 3.0. Each revision boosts the speed per lane.
x16 signifies the number of lanes per slot. Your graphics card is a PCIe 2.1 card which uses all 16 lanes. The x1 slot is slower, and can be used for other types of cards.
EDIT: I just saw the last part of your question. Yes, you can use the PCIe 2.1 card in the old PCIe 1.0 slot because it has the correct number of lanes (16). But it will not perform at its full potential, because PCIe 1.x carries 250MBps/lane and PCIe 2.x carries 500MBps/lane.