Decrypting in Mass Effect is different on consoles in comparison to PC. I will cover the different variations and provide some gameplay videos on how each of the mechanics work below.
Consoles (Playstation 3 and XBox 360)
The visual 'feel' of decryption differs across the consoles but the underlying mechanic is the same. On the console versions of Mass Effect, when you attempt to decrypt you are presented with something similar to this:
Obviously this one is from the XBox version of the game (based on the button names). The object here is that as parts of the puzzle highlight, you have a small window of opportunity to press the corresponding button on the control pad (square, circle, triangle or cross for PS3). Pressing the wrong button will fail the challenge.
Depending on the difficulty you will be required to perform a sequence of between three and seven button presses.
Here is a quick video demonstrating this mechanic in action, the reaction time required is reasonably fast and I feel this is definitely less forgiving than the original PC release:
For the PC, decrypting presents you with a screen that looks like this:
This is a timed mini-game with the time remaining showing in the middle (in the instance of the above screenshot there are 12.10 seconds remaining on the timer) where you must get the arrow from the outer circle to the centre without hitting any of the blocks. The red blocks will rotate either clockwise or anticlockwise, each of the concentric circles can rotate independently of each other and hitting any of the blocks will throw your arrow back to the outer ring where you started.
This video shows the mechanic in action:
Getting to the centre circle will complete the decryption process.