My answer on Quora:

The credibility of a central bank plays a strong role in its ability to achieve its objectives, which often includes price stability (i.e. stable inflation around a desired target). If a central bank lacks credibility, investors and firms won’t have much confidence in the central banks ability to keep inflation around its target, so the central bank may need to go above and beyond to make an impact on pricing decisions (and inflation). On the other hand, if the central bank is considered credible, then sometimes all it takes is for it to simply announce what it will do to bring inflation to its target to influence pricing decisions.

• With credibility, people believe the central bank when it says it will act and, most importantly, that it will be successful if it does it (i.e that its monetary policy is effective and can filter through the banking system and influence borrowing/lending, for example).
• Without credibility, the central bank not only needs to follow through on what it says it will do, but might need to act stronger than it otherwise would need to in order to make an impact.

The latter example is costlier.

Credibility is important because it influences inflation expectations. If people expect prices to be higher tomorrow, consumers will buy now but investors might be reluctant to invest now since the value of money (more specifically, a specific unit of currency) decreases over time. Price stability is therefore an important objective for a central bank because it provides an anchor for inflation expectations and the spending/investing decisions of firms and households.

Central banks can try to manage inflation by setting key interest rates and controlling the money supply. There are many mechanisms within each of those channels, like the overnight lending rate between banks, reserve requirements, and printing money.