No, actually the video card can run in a PCI-Express x 4 slot. I just bought an Asus A8N5X motherboard (I'll bet that's what you got) and in the manual you can put a PCI-Express x16 video card in the PCI-Express x 4 slot. It just doesn't run as fast.
And, most video cards DO have something from the power supply that plugs into them. What kind of video card are you using? If it is above something like an ATi 9700 or GeForce 6000 series, then you probably should have something going from your power supply into your card. And since you're using a PCI-Express x 16 slot you can't really get any old cards for it, so I bet you do need to plug something in. My card came with a 6 pin adapter to 2 molex connectors (eVGA 7800GT).