Carbon dioxide dissolves in water /beer. The colder the temperature, the more dissolves at a given pressure. The higher the pressure, the more dissolves at a given temperature. It's not a linear thing, so you probably want to look at a table to work that part out.
The gas dissolves in the beer slowly. It's not a poof done sort of thing. The faster you want to get things done the more pressure you need (at a fixed temperature). Kegs explode at about 100 psi. Pressure relief valves go off around 60 and may leak above 30. There are practical limits on what you can do.
Different styles of beer call for different levels of carbonation. If you serve me a liter of lagger and there's no foam on it - I'm not happy. If I sit down for a nice cask ale and it's got three inches of foam on it - not a good thing.
Simple answer - you want to cool it to force carbonate it. How long and at what temperature / pressure depends on the type of beer.