They are in series as you must control the current supplied to each LED and the only way you can control that many is to operate them in series, suggesting that they may require a constant current drive capable of operating at something of the order of 30 to 50 Volts.
If you can figure out just how many LEDs are in each chain, you will have an idea of what is required. They will probably use about 20 mA (but this is only a guess) per chain. It is not entirely obvious from those photos exactly what sort of LED modules they are and without a part number (and even then only with luck) we do not know their actual specification.
shurkan2:
I would lie to power an LED-strip of an old external computer-monitor with an Arduino.
Sorry, but "powering with an Arduino" is not a meaningful concept. An Arduino is not, and does not in any way resemble a power supply. It is indeed, a microcontroller, not a power supply.
Clearly, the circuit board on the monitor already contains the necessary power supply components and if it is working, is the most sensible way to operate the LEDs. However it is probably a trifle too cumbersome to use merely to light the LED bar.
I am reasonably sure from looking at your pictures that there are just two chains of LEDs - four traces running along the LED bar. I would suspect that alternate LEDs belong to the two alternate chains. At a reasonable guess, the LEDs drop about 3 V each (this may be wrong, each LED module may have more than one LED in series), so if you count the total number on the strip and multiply by three, then half of that is an approximation of the voltage each chain requires, so if you find a power supply that can supply about 10 Volts more and is either variable so you can gradually wind up the voltage to one chain until they light and measure what that voltage is, or connect them in series with a 1k resistor of at least one watt rating to that power supply and measure what the voltage is across the actual chain when they light.
Knowing what voltage they require, you ill need a constant-current driver which can supply the 10 V more at a suitable current - I suspect it may actually be 300 mA or so as I presume the total power is intended to be several watts. Most sensible would be to measure the current drawn and the voltage across them when powered by the original board - which would require some very careful fiddling with the connections.
Quite frankly, considering the fiddling involved, I have to ask what you actually hope to achieve here?