Since the early 1980s, most central banks have communicated their operation of monetary policy by setting a policy nominal interest rate. For example, the Fed in the US communicates the stance of monetary policy through its target for the federal funds rate (FFR); the Bank of England communicates the stance of monetary policy through its direct control of the official bank rate. (This in contrast to e.g. Milton Friedman's famous proposal for a rule for the growth rate of the monetary base.)
Since all interest rates in an economy are linked by arbitrage pressure, the Fed by controlling the FFR can affect more general interest rates facing consumers and businesses – such as interest rates on bank deposits. The consensus understanding of monetary policy is that by controlling the current and expected future path of these interest rates, central banks can influence economic activity: GDP, inflation, unemployment, and other metrics.
However, it has long been believed that central banks cannot push their policy rates much below zero. This is because of a no-arbitrage relation with physical cash. By holding physical cash, you can guarantee yourself a 0% nominal return: if you have 100 dollars in cash in your wallet today, you can ensure that you'll have 100 nominal dollars in your wallet next year, therefore guaranteeing a 0% nominal return. Thus, if the central bank tried to set its policy rate to cause the nominal bank deposit rate to be (say) -20%, then most everyone would pull their money out of the banking system and hold cash instead, to obtain the higher interest rate. (This is essentially a form of Gresham's Law: central banks peg physical cash and bank reserves at a 1:1 rate; but the rate of return on the two need not be the same.)
This zero lower bound (ZLB) on nominal interest rates however is not a hard limit: because of the costs and inconvenience of storing cash and preventing theft, individuals and firms are willing to accept a somewhat negative nominal interest rate. Indeed, in Europe, the policy rate has been slightly negative since 2014, and this has spilled over to interest rates facing households, such as mortgage rates in Denmark.
In the United States, unlike the ECB in the Eurozone, the Fed has been unwilling to target a policy rate even modestly below zero. Various Fed officials have argued (among other things) that they lack the legal authority to do so; or that benefits would not be large enough to outweigh some perceived risks.
A history of the Federal Funds rate can be found at FRED