I hope that all the previous wrong connection did not damage your motherboard yet. First connection, connecting the M/B 24 pin cable from new PSU to the female 24 pin adapter.  Then connecting the 8 pin end of the adapter to the motherboard 8 pin socket with marking ATX_SYS. Second connection, using the cable from your latest post and connecting "ESP 8pin/ATX 4pin" to connector on PSU.  The other end of cable has 4 + 4 pin with marking connects to motherboard should be connected to 4 pin socket next to the CPU to provide power to the processor.  Test each 4pin and you will see that only one will fit into that socket.  Note that the locking clip on 4pin connector should line up with locking groove on CPU 4pin socket. 
Does anyone have experience with installing a 3rd party PSU on these machines? I have a video card that requires a more powerful PSU than the stock 365w PSU that is in it right now. I bought an adapter but i get orange flashing lights on the power button. I did not wait long enough to see the code given by the flashing lights as i was worried i would fry the motherboard or components. Then I tried the "ESP 8 Pin/ATX 4pin" cable that came with the new PSU the PC would not power on at all. I circled the ESP 8 Pin/ATX 4pin part of the new PSU in the picture below.  Any help is much appreciated.   Here is the adapter I used. [URL] [IMAGE]     Here is the PSU I bought [URL]   [IMAGE]  
3rd Party PSU for Precision Tower 3620