The first possibility to take a screenshot from your Samsung Galaxy S3 is with the option “palm swipe to capture”. This option is enabled per default under settings / motion. But how does palm swipe to capture work? I tried it a couple of times and I did not figure out but finally I got it:

But your hand upright with your little finger on the display and move your hand from right to left or vice versa. Afterwards the typical screenshot sound comes up.

Here you will find a short movie:

Second possibility is to push the power and home button at the same time and to wait a second. Here is also a short video: