APY - annual percentage yield, called AER in the UK and Europe - is what a rate actually earns over a year once compounding is included. It is the only fair way to compare two accounts quoting different compounding frequencies.
The nominal rate is the headline figure. APY is what you actually earn once interest starts earning interest during the year. A 5% nominal rate compounded monthly gives an APY of 5.1162%; compounded daily it gives 5.1267%.
The gap widens as the rate rises. At 5% the difference is about a tenth of a percentage point; at 15% it is closer to a full point.
These three get used loosely and mean genuinely different things, which matters when comparing products across markets.
APY (annual percentage yield, US) and AER (annual equivalent rate, UK and EU) are the same idea under different names: what a deposit actually earns in a year once compounding is counted. Both are what you want when comparing savings accounts, because both fold the compounding frequency in.
APR (annual percentage rate) is different and is used for borrowing. It represents the cost of credit, and in most jurisdictions it must include certain fees as well as interest. Critically, APR is usually quoted as a nominal rate that does not compound the way APY does, so for the same headline number an APR and an APY are not directly comparable. Australia uses a comparison rate for a similar purpose on loans, and the UK uses APRC for mortgages.
The short version: compare savings with APY or AER, compare loans with APR, comparison rate or APRC, and never compare a savings APY against a loan APR as though they were the same measure.
Because two accounts can quote the same nominal rate and pay different amounts. If one compounds daily and the other annually, the first earns more. APY removes that difference so you are comparing like with like - which is exactly why regulators require it to be published.
APY = (1 + r/n)n - 1, where r is the nominal annual rate as a decimal and n the number of compounding periods per year. Reversing it gives the nominal rate implied by a quoted APY, which is useful when a provider advertises the APY but your own projection needs the nominal figure.