Money market accounts are almost the same thing but the main difference between a Money market account and a savings account is that with a money market account you can withdraw funds from the money market savings account up to 3 to 6 times per month and they pay higher interest rates on your money than a regular savings account.
A money market savings account also sometime has a higher minimum balance requirement than a regular savings account.
As for a regular savings account they sometimes pay less interest, do not allow withdraws for a certain period of time and you can usually start the regular savings account with less money and don't have a minimum balance requirement.
I have and prefer a Money Market Savings account because it pays higher interest than a regular savings account.