I don't know why I'm posting this. But it seems that there are people who don't have one and probably don't understand what it is.
Simply put, a savings account is a place to save your money and you will earn monthly interest on it. You can open a savings account at any bank. In the United States, almost all the banks are FDIC insured. That means, if something were to happen to your account, your account is insured up to $100,000. So if you had $20,000 in the savings account and someone stole it, you are insured up to $20,000.
Keep in mind, interest rate on your savings account are subject to change. I remember I was getting 5% on my savings account in the 1980s. In the late 1990s to the time I write this, I now get less than 1% on my savings. Then I came across online savings such as EmigrantDirect, HSBC Direct, Citibank e-savings, and so on that gives 4.50% to 5.10% on their savings accounts. These too are also FDIC insured.
So how you open a savings account? Go to a bank and ask to open one up. They will tell you what you need to open one. I would pay special attention to their fees (if any), the minimum balance requirement, and the interest rate they offer.
What can you do with your savings account? You can deposit and withdraw money from it at anytime. Though, you won't be able to pay your bills with cash (and no legit company is going to accept cash as a method of payment). You need a checking account to pay your bills.
How much should you have in there? I wouldn't keep too much in there since they don't have a great return on them. They are good for short-term uses such as going on vacation or buying a home or for emergencies. Though, I would recommend using money market funds as your emergency fund. They tend to perform slightly better than what you get in your savings account.
If you are going to save for long term such as retirement, you need to invest. This is how people become wealthy when they retire. They invest early and stick with it for the long term. I suggest investing into mutual funds. Some say you should invest in no load funds. I say, it doesn't matter if the fund is a no-load or load fund. Both will get the job done.