The main difference between the two is how much interest you earn.
With a money market account, you could earn a slightly higher rate of interest than with a regular savings account. Money market accounts may also feature tiered interest rates, with potentially higher interest rates for larger balances.
A checking account is primarily designed as a tool for receiving deposits, such as your paycheck. You can then use the money to pay bills and make purchases, either with checks, via Online or Mobile Banking, or with a debit card.
A savings account is where you put money that you hope to save. Savings accounts are typically used to save for large purchases, vacations, or an emergency fund.