FTTC stands for Fibre To The Cabinet and is often just referred to as 'fibre', at least in the UK. This can be a bit misleading however, as many people will expect 'fibre' to be a fibre link into the home or the office but this is not the case with FTTC.
As the name implies, the fibre link is between the local exchange and the green telecom cabinet at the end of your street or the side of the road. The connection between the telecom cabinet and the home or office uses the existing copper telephone lines. The performance of of these copper lines for data transmission decreases with distance so even if you're on a 'fibre' product, if you're a long way from the street cabinet your speeds could still be quite low.
In the UK, FTTC connections fall under the term Superfast broadband which covers download speeds in excess of 30Mbit/s.
Crucially, when it comes to ordering FTTC, this needs to be ordered on top of an existing traditional analogue telephone line. Over the coming years, FTTC is going to be superseded with the new SoGEA product.