Example 1

This is an example of applying the Light Chasing algorithm. In this case, the puzzle is solved after the first run through. In general this won't happen! You could try following this puzzle through on my javascript Lights Out. It will make it a whole lot clearer!

This is where we start. We see that the lights in the 1st, 3rd and 4th positions on the top row are on. The Light Chansing algorithm says that therefore we should press buttons 1,2 and 4 on the second row. Do this
Now we have lights 1 and 5 on the second row lit. Therefore we should push buttons 1 and 5 on the third row.
This time it's lights 2 and 3 on the third row - so once again press the corresponding buttons on the fourth row. Simple isn't it?
Almost there. if you've played with Lights Out much, you probably recognise this position! Following the algorithm, we have to push buttons 4 and 5 on the fifth row.
And that's it! Lights Out! Congratulations :-)