An equivalent interest rate is the rate which achieves the same total proceeds as another given interest rate (the nominal rate), but assuming a different frequency of compounding.
An effective interest rate is an equivalent interest rate, where the frequency of compounding is annual (i.e. 365 days).
A continuously compounded interest rate is an equivalent rate, where the frequency of compounding is infinite (i.e. the period of compounding is infinitesimally short).
Suppose bank A quotes 10% per annum (the nominal rate) for a 9-month (270 days) deposit, with all the interest paid at the end of 9 ...