Could i use a car fuse or does it need to be a glass one
Ugh! Car fuses are horrible! Use a 20mm glass fuse. As to the comments about breaking high current, probably does not matter at 5V or 12V, but the simplest solution to extinguishing the arc is to use a HRC (high rupture capacity) fuse. HRC fuses are typically filled with (I think quartz) sand, which quenches the arc. In the UK common mains plug fuses are of this type. Look for HRC in the specification.
I am assuming that in a short circuit situation the power supply draws more amperage than what the circuit can handle and that is what blows the fuse, is that correct.
The power supply does not draw current on its output, it supplies current. You might or might not be correct, depending on the power supply. A lot of power supplied use foldback current limiting, which means that when you short the output instead of supplying their maximum current the current drops to a very low level until the short is removed.
I understand if I separate the strip into sections i can fuse each section with a 5 amp fuse but I need to think about how to hide the wires.
Maybe I missed it somewhere so apologies if this has been said already but to be clear the fuses need to be close to the PSU, not close to the LEDs.
Quite frankly, I don't see how you can feel comfortable telling this to a newbie who would have no way of knowing (without the AWG) that 1 mm^2 is 17 guage (1.04 to be exact)(AWG) which is in between 16 and 18 guuage and which almost nobody sells. This is useless information for a newbie.
How strange. 1mm^2 wire is easily available in the UK, I have 2 rolls of it right next to me.
Stebod, using thicker wire than calculated is not a problem, just a bit more awkward to work with.