A savings account is insured by the FDIC. If the bank goes out of business, your money in the account is safe, up to 250,000.
However, most of your money is not really "held" in the bank. The bank turns around and lends about 90% it to other people so they can earn interest and investment income. Normally savings account pay very low interest rates, not even enough to keep up with inflation.
A money market account is an investment account and it is not insured. You could technically lose everything in it. However, this has never happened in the history of money market accounts, due to the very safe types of investments (high volume of very short-term treasuries and commercial paper). Additionally, for every dollar you put in, there is an actual dollar backing it in the account.
Normally money market accounts pay higher interest than savings accounts, however, in this day and age this may not be happening currently. You need to confirm the rates.
Both Money Markets and Savings accounts are generally used to store "emergency funds" These would be funds that are easily accessible and intended to retain their value of the short term, so you have access to cash, if you need it.