The difference between an old non-condensing boiler and a new condensing model is the amount of useable heat it produces from the fuel it burns. Old boilers usually have a pilot light, which is burning all the time, whether the boiler is on or not. A non-condensing boiler could be as little as 60 per cent efficient, meaning for every pound spent on gas, 40p is lost.
A condensing boiler is much more efficient. It has a much larger heat exchanger, which extracts over 90 per cent of the heat from the fuel it burns, making it much more cost effective to run. It extracts the heat from the flue gases, which would otherwise be lost to the atmosphere, and recycles the heat back into the heating system. The water vapour in the flue gases can be so cool that it condenses (hence the name!), and is taken away to a waste pipe through a condensate pipe.
Condensing boilers do not need a pilot light burning all the time, because they fire up when there is a demand for heat, using an electric spark.