To say that something is "compounded quarterly" is to say that it is compounded four times a year. An example of something that may be compounded quarterly is the interest rate on a bank account.
The interest rate the bank provides is typically an annual or yearly interest rate. When the interest rate on the bank account is compounded quarterly, the calculated interest for each quarter after the first quarter takes into account a new principal amount that includes the interest from the previous quarter or quarters. Basic interest differs from compound interest in that it is paid only once.