PCIe (express) is different and is in no way compatible with PCI.
the x16/x4/x1 refers to the amount of lanes the slot has (and max throughput)
the PCIe 3.0 refers to the revision of the PCIe slot. A PCIe 3 video card will work in a PCIe 2 slot.
PCIe 2.0 is the same throughout. There is no "normal PCIe 2.0". The "2.0" simply refers to the standards version which the slot was designed to conform with (primarily data throughput).
PCIe 2.0 x16 (Blue) means the long blue slot on your motherboard runs at the full throughput of the slot's configuration. Think of it as a sixteen lane highway.
PCIe 2.0 x16 (x4 mode, black) means the long black slot on your motherboard only runs at 1/4 the throughput of the slot's configuration. Think of it as a sixteen lane highway with twelve lanes closed for construction.