Hi Jipsta and welcome to the forum.
A remap is a way of reprogramming the computer chip that controls the engine so it produces more power (horsepower) and torque (pound foot) than as standard. It involves altering things like the turbo boost pressure, the fueling and the rev limiter. The standard programme as it comes from the factory could be thought of as a bit of a compromise to cover all the ways that an S3 might be driven. For example, the standard programme might target giving good MPG at the speeds where MPG is measured for sales purposes. Or giving CO2 emmissions for tax purpose.
The S3, as a turbo engined car, responds very well to a rechip - you will get somewhere between 250 and 260bhp compared to the 210bhp as 'standard' with your car (I say 'standard' as VW group cars are often over book figure anyway). The figure varies upon which chip you have, and which rolling road is used to measure the power (some are more optimistic than others).
So for somewhere around £400 you will make your car noticeably faster and with improved pickup in any given gear (for example, you won't need to change down a gear to overtake that lorry).
What are the downsides? There are some you need to be aware of:
1. You must tell your insurance company - it will put your premium up, but some insurers are more friendly than others.
2. If you get used to your extra power, then your MPG will suffer. Having said that, quite a few people notice that MPG actually improves on long motorway trips (less gear changes are required thanks to a better spread of power)
3. Having more power means that you tend to drive a bit faster which might emphasise some of the other parts of the car - e.g. brakes, suspension etc which you might want to upgrade as well.
4. If your new re-map gives you a bit jump in torque (this varies from supplier to supplier and car to car), this can cause premature wear in the clutch. But again, this depends upon how you drive and lots of members have had remapped cars for years with no problems.
Hope that helps you!