Where you put savings depends on how much you have and what you plan to do with it.
For only a small amount of money; with less than $200 to $500 or so, you have got problems. Hardly anyone wants your money. Banks might let kids have a small, free savings account. But as an adult, you will probably pay service charges, and they might exceed the interest you earn. Some banks will not pay any interest at all. So instead of growing, your savings account would gradually shrink. One solution: Add that money to your checking account and do not spend it. The extra $500 might give you enough to qualify for interest paying checking. Another solution: try a credit union, which is more likely to accept small savings account without charge.
A statement savings account might pay interest on small deposits ($100 to $500) and still charge no fees. You do not get a passbook. Instead, you get a statement every month. I like the convenience of statement savings. You do not have to remember where you put the passbook and can usually handle the account through an ATM. At this writing, you earn around 2.5 percent instead. You can take out your money at any time.
A passbook account might require a higher minimum balance. Then again, it might not. Everything depends on the bank. Deposits and withdrawals are recorded in the passbook; interest rates run in the area of 2.5 percent.
For larger amounts of money; a money market deposit account pays a variable rate of interest and I do mean variable. The bank has the right to change it at will. In high rate periods, it pays about the same. You get unlimited deposits and withdrawals, although withdrawals might have to exceed a certain minimum amount. You are allowed up to 6 pre-authorized transactions a month, only 3 of which can be checks to third parties (the others can be automatic bill paying or checks written to yourself). You will be charged monthly fees if your deposit falls too low. Money market accounts are a good place for money you are likely to need pretty soon, like rainy day money, money waiting to be invested somewhere else, money you will need to live on for the next 6 months and earmarked money, like a tuition payment due in a few weeks.