The instructions I've been following say to leave the dough in the fridge for 24 hours. I don't know much about bread - but could someone explain that to me?
I thought that being in the fridge would stop or slow the yeast from working - could I just leave the dough on the counter for less time and get the same/similar results?
Yes, the yeast will slow down its activity in the fridge, allowing more complex flavors to develop in the dough from a cold fermentation.
It's all personal preference.
You can certainly make pizza with a one-day dough.
There's a number of variables in the ingredients you use that could affect flavor and texture.