There are different types of overflow that you may potentially find in your home. I will go through each one in turn.
1. Overflow from your toilet cistern
These days most toilets incorporate an internal overflow. So, instead of a separate pipe carrying excess water away, it drains directly into the toilet bowl. You can of course have your flush valve replaced to update your toilet so it has an internal overflow. The internal overflow is a great innovation as it does away with the extra overflow pipework and is also more reliable (overflow pipework can leak and still cause damage to your property).
2. Overflow from your cold water header tank.
If your hot water is supplied by means of a hot water cylinder (usually in the airing cupboard) then you will have a header tank (often in the loft) that keeps your cylinder topped up with fresh water. This will also have a ball valve or fill valve (just like your toilet). If this valve fails then the excess water will be carried away by an overflow pipe and terminate outside your home.
3. Overflow from your central heating feed and expansion tank.
4. Overflow from a combination boiler (pressure relief valve)
If you have a combi boiler then it will have what is known as a pressure relief valve. If your system builds up to much pressure then this relief valve will open and allow water to escape through a pipe and terminate safely outside your property (this escaping water could be hot so needs to terminate in a location that could not scald someone). Pressure relief pipework will be made from copper. Sometimes your pressure relief valve might start 'letting by' resulting in a drip from your external pressure relief pipe. If this is the case, you will notice that your boiler loses pressure and needs topping up on a regular basis.