I know what your problem is. You wired it wrong to the arduino. The diagram is a little tricky on how it is wired. Try this and get back to me. Your rows should be cathodes and the columns should be anodes. I took the liberty and drew this diagram for you.
@jaysee - common anode. But I don't know what's the difference.
@Matt - I didn'u use ANY resistors. Should I use here?
@FallenDemon - my matrix led has goldpin on the opposite walls. I don't know where is first (matrix) pin which I should connect to 9 arduino pin. Is it difference where I start?
Your schema has 16 pins (in Arduino) - but in standard Arduino (eg. Duemilanove) are 15 ports (0-14). Where will I connect a sixteenth pin?
Anything past pin 13 is going to be analog. Pin 14 would be A0. 15 would be A1 and so on. From the looks of the video it is definitely a wiring issue. I built my 8X8 matrix from scratch using my schematic and it works wonderfully. You definitley should use resistors. 8 of them to be exact. Try reversing the wiring. On your matrix where you wired pins 1-8 use 9-16 instead. Good luck.
The schematic you posted is for common cathode (i.e. ROW = CATHODE) displays. If you use a common anode (i.e. ROW = ANODE) display you have to switch the connections (13 <-> 9, 3 <-> 14,...). See the attached image. And as Matt wrote, you really should use resistors to limit the current (R = 470 Ohms should be OK).