Some very quick explanation of the theory here:
This is the input stage of the power supply. Its job is safety, surge protection, removing some noise and transients, rectifying the AC current to DC, and smoothing out that DC with a big capacitor.
The blue circle should be the connection to the power cable that plugs into the wall.
The purple circle is a fuse that cuts off current from the blue circle if things go wrong. It may have thrown, you should test for continuity between these points. This might be what has "sparked" when the device failed.
The red circle is a full-bridge rectifier. You can look up how these work. It's basically a package of four power diodes which turns AC into noisy DC.
The yellow circle is the largest capacitor on the board, the failed one. The job of the largest cap is almost always to smooth out the DC created by the rectifier, the red circle. The big fat circuit trace connected to its cathode (the curved line on the symbol) is your DC ground plane. That will be your "negative" reference for the rest of the power supply. The other side is your main DC positive.
Post too long. Click here to view the full text.