There are two popular ways of carbonating a beer - bottle conditioning and force carbonating. In this video, we cover bottle conditioning.
Bottle conditioning requires less equipment - all you need to do is present more sugar to the yeast in the beer and seal it in a container. Since yeast produces CO2 as it consumes sugar and the CO2 in a sealed container has nowhere to go, the excess pressure is then reabsorbed into the beer making it carbonated.