The central bank sets the interest rate at which it lends to other banks. This acts as a control upon the average interest rates.
What would happen without the central bank? Hard to say, because interest is almost always influenced by some sort of state intervention in the matter, i.e. before (and for a bit after) the Federal Reserve was formed in the US, the country was on a state-mandated gold standard, which obviously would be have different effects to an entirely paper based one like we have now.