I suppose it depends on what case you have. Your third picture shows the connections for the case/front panel.
PWRSW is your Power Switch.
PLED is the Power LED.
IDE_LED is the drive activity light
RESET is the reset switch.
Now if we know what the cables from the case are, problem solved! What is the case? Are those pictures connectors from the case? They look like hard-drive power connectors to me, but I'm not good at guessing from pictures.
Delluser1 says that it's USB, so let's hope so. Since you already plugged a usb cable into the connectors, I'm going to assume that you can find another USB connector for it.
For the front-of-the case connector, how many pins does it have and what is the color order? It may not fit your connectors - your system's power button / light button pin pattern is light-light-empty-emtpy-power-power, and that doesn't look like a six-pin plug.
What model is the case, and do you have the case manual?
I suppose I could have put a question mark at the end of my last post, not knowing what case it was in.
Not a good photographer, but you can see the same pinout on this dual port usb connector as in the OP's photo
That said, the OP's connector being in an Emachine case, could be anything