Ok. There are basically two types of system - vented and unvented. Your current system is vented and relies on gravity to supply hot water (HW) to your taps. Your central heating(CH) (the smaller tank) may be pumped or an older style of gravity circulation.
A combi will mean that your CH will most probably (although not always) become a closed and pressurised system which will mean that you lose the smaller (FE feed and expansion)tank in the loft and will be fully pumped and require you to keep an eye on the pressure - a simple task. The HW will be supplied at mains pressure, which is usually better for showers, but not so good for running baths since it will be able to provide water at about 13 to 15 litres per minute. The advantages are that it will mean you lose the tanks in the loft and there is only one place where water is kept and supplied hot.
The other type of pressurised system is one a bit like you currently have in that the cylinder is replaced by a pressurised hot water vessel, which again will provide hot water at mains, but will provide a faster bath filling time. The CH can be as it is now, or sealed as with the combi, and tank(s) in the loft will become obsolete. The disadvantage is the cost of setting up - you need a boiler and a hot water storage vessel ("cylinder").
Whatever you choose, the boiler will be condensing, and at Sedbuk A (90%) or B(86%) efficiency - most likely A, and running costs will be drastically reduced.
Either way, you may be able to re-use the existing pipework, depending what's there, and all but one of the radiators should have TRV's fitted, which means that rooms will have their own cut out when they get to temperature, which again adds to efficiency.
It's a start - what else do you need to know?
