The dollar, the euro, sterling, the yen, the Swiss franc and the Australian, Canadian and New Zealand dollars are treated as free floaters, though several have intervened at extremes. The defining feature is that no level is defended as policy; the exchange rate is an outcome, not a target.
Floating gives the central bank a free hand on rates and lets the currency absorb shocks. A commodity exporter whose export price collapses sees its currency fall, which cushions local producers, a mechanism covered in terms-of-trade and commodity-currency.
The cost is volatility, and the reason forex is tradable at all. Floating rates move on interest-rate-differential expectations, growth, and risk appetite, rather than on an announced schedule.
Example: since floating rates became the norm after nixon-shock, moves of 10% or more in a major pair within a year have been ordinary rather than exceptional.
Related: managed-float, nixon-shock, bretton-woods, terms-of-trade