My Samsung Galaxy S6 fast charging cable has broken and I was wondering whether I needed to buy a cable specifically for fast charging, or just a regular media cable.
The fast charging plug socket isn't broken, it's just the cable.
Fast charging is not related to the cable - the circuitry is in the adapter and device. That said you need to buy a cable that can carry higher amperage , allowing fast charging speed ( cable per se doesn't increase the speed but because it permits more power, speed is increased)
Thicker the cable, better it is ( thicker cables have lower AWG number )
Go in for a cable like Tronsmart cables at a price comparable with budget cables. These are 20 AWG thickness for charging (provide best charging) 28 AWG for data (thinner thickness doesn't affect data transfer speeds)
For technical details of a good quality cable see my answer here
I am not affiliated with the product- a very satisfied user
You'll want a quality cable, but the important part is the fast charging adapter itself, and if that's not broken or anything then you should be able to use any good quality cable.