There are two main control modes in the whole game: Low Profile and High Profile. The first one is any activity, which does not provoke other people's negative reactions. High Profile is basically opposite to Low Profile and activating it (using the right trigger) can make an assassination visually far more interesting but is sure to bring nearby guards to attention and result in an all-out fight or a chase through the city.
Using Low Profile for the same activity will not bring as much attention and your mission can be finished more discretely.
Here is a list of basic activities and buttons for both Xbox 360 and PS3 respectively:
High Profile - push a right trigger/R1
Left analogue stick
Right analogue stick
moving a camera
Button under the left analogue/L2
adjusting a camera
High Profile actions