Savings accounts are the most popular way to traditionally save money. Typically they are not fancy, just an account that usually pays a fixed interest rate. Interest it usually paid to the account at the end of the month.
The great thing about savings accounts is that there is virtually no risk – unless you have a considerable sum of money in the bank. Recently the FDIC (Federal Deposit Insurance Corporation) has raised the amount of insurance per depositor from $100,000 to $250,000 until December 31, 2013. At which time it could be extended or revert back to $100,000. Even in the event of bank failure, you are guaranteed to get your money back up to the limit of FDIC insurance. If you have more money in your account than the FDIC limit, you should consider opening another account(s) at another bank so it will also be insured.
Also, in a savings account, your money is accessible to you. It is not tied up in stock or some other asset that could take time to actually convert to cash. If you need money, you can transfer money right to a checking account or simply drive down to your local bank and withdrawal the funds.
The downside to a savings account is that you are not going to earn much interest. It usually pays less than bonds, CDs, etc. The reason you are not earning much interest is that you have complete access to your money when you need it, so that is trade off. However, you can usually find some banks on the internet that will pay a little more interest than traditional banks. Some people are turned off by this because you have no interaction with people and there is usually not a branch you actually walk into if you need it. If you are okay with this, then an online savings account may be for you. When you open an online account, always make sure it is FDIC insured, and you can feel safer about your money being there.