Simple interest is calculated only on the original amount, never on interest already earned. That makes it easier to work out than compound interest - and, over any real length of time, considerably less rewarding.
The formula is I = P × r × t - principal multiplied by the annual rate, multiplied by the number of years. Interest is worked out on the original sum every year, and never on interest already added.
So $10,000 at 5% earns exactly $500 every year, no matter how long you leave it. After 20 years that is $10,000 of interest, giving a total of $20,000.
Compound interest is paid on interest already earned, so the balance grows faster each year. The same $10,000 at 5% for 20 years reaches $26,532.98 with annual compounding, against $20,000 with simple interest - a difference of $6,532.98, entirely from interest earning interest.
Almost all real savings accounts, mortgages and credit cards use compound interest. Simple interest turns up mainly in some car and personal loans, certain bonds, and short-term lending. If you are not sure which applies, assume compound and use the compound interest calculator instead.
Most often on fixed-term loans where the interest is calculated once at the start and divided across the repayments - some auto finance and personal loans work this way. It also appears in Treasury bills and other discount instruments. For anything held over years, compound interest is the norm.