Carbonated drinks are so called because they contain carbon dioxide gas dissolved in the liquid under pressure. When you release the cap, you release the pressure, and some of the gas is released as bubbles, making the drink fizz. However, a lot of the gas is still dissolved in the liquid, keeping it fizzy as you drink it. Carbonated soft drinks are referred to 'carbonated', 'aerated', 'fizzy' or 'sparkling'. If the drink isn't kept under pressure, over time all of the carbon dioxide will eventually escape and the drink becomes 'flat'.
A non-carbonated drink simply is one that hasn't got any carbon dioxide in it. These are referred to as 'still' soft drinks, or, in the case of bottled water, 'natural'.