Please excuse my stupid, but does this fit this? Gigabyte GA-Z170-HD3-DDR3 + Intel Core i3-4170

No. If you want to use that motherboard you need to use a 6XXX (Skylake) series intel processor where the processor you linked is 4XXX (Haswell). If you would like to use that processor, you would need to use a Haswell motherboard (HorZ9X).
 
Thankyou Gilbadon and Anarkie. that saved me some messing around. 🙂

I'm not sure whether to choose a different motherboard or a different processor, they both have glowing reviews, but I guess it doesn't really matter all that much, it's a bit of a coin-flip. That alternative processor looks to be cheaper than normal, so that's tempting.