GTX 770 does not require 2 PCI slots. It has one connector, that is PCIe x16. (the little part to the left of the connector is for power and the rest is for data connections).
In your image, all the black ones (except for RAM slots) are PCI-Express (PCIe) slots. Longest ones are x16, smallest one is x1 and the one you red-circled is the x4 slot of PCIe. You can install your graphics card to any of those (even though it has x16 lanes), your graphics card will work. However, for the best performance you have to install your graphics card to a x16 slot.
More about PCIe: you can install PCIe of any length (x1, x4, x8, x16) to a slot of any length. Means that you can install a x1 device to a x8 slot or x16 slot to a x4 slot. However, you should choose the slot according to the length of your device for best performance.
To clarify things: you provided an empty slot image and asking for what is in there. Can be anything with a PCIe connector. You should check back of the chassis for connections, maybe you can identify what hardware is installed on that slot. Furthermore you can change its position. If you have the other PCIe x16 slot empty, you can install the PCIe x4 device there, or even to the x1. So that you can get enough room for your new GPU.
This question, considering room for relatively larger GPUs, might interest you as well.