A combination boiler combines the functional requirements of a central heating boiler and an instantaneous water heater. You may remember the old style of Ascot water heater that used to sit over the sink - it heated the water as required. A combi does that, and hence has no need for a hot water storage tank.
Advantages are cost, no need for hot water tank and hot water is always available. Disadvantages are the hot water flow rate can be slower and this is particualry true in winter when the CH is running - the very tiome more of the family want hot showers. Also they can tend to scale up quicker thought there are ways to combat this successfully.
New installed boilers have to be 'condensing' type these days - these have higher efficiency but cost more than older-style. Perhaps �2-2.5k - depends what has to be done to the flue arrangement.