A combi boiler provides CH and domestic hot water (DHW). The CH is conventional but the DHW works by running the cold water from the cold water mains inlet across a heat exchanger (which is heated by the boiler) AS the hot water is demanded by turning on the taps. It thus has no hot water storage tank (it doesn't need to have one). A combi boiler differs from a 'normal' boiler, which heats the hot water in a separate storage tank, often in the airing cupboard.
A condensing boiler is a method of combustion of either gas or oil that extracts more of the available energy from the fuel. Condensing boilers may be combis or they may be 'normal' types. Since about 2005, Government policy has forced new installations to have a condensing boiler type fitted. The simple explanation of how they work is they have an additional heat exchanger in the outlet flue pipe that cools the outgoing combustion gases to a lower temperature than that achieved by non-condensers. The additional heat extracted makes them more efficient - hence the Government desire to introduce them compulsorily.
In the early days of introduction, manufacturers rushed to get new products to market and there were reliability issues with some makes. The worst of that is probably over now - though some makes are still better than others.
If you are installing a new boiler in a new property, it will have to be a condensing type.