Decide what you are protecting them against. Options include (but are not limited to): Accidental disclosure, disclosure by you, disclosure in transmission, disclosure due to code error, disclosure due to physical theft of hardware, etc.
If this is a web application, and each user is storing his/her own set of passwords, then you might encrypt these passwords with their login password to your application. If this is an application that each user installs separately, and which keeps its own local database, you could have an optional master password (like Firefox does).
If you are just ensuring that the data is safe if the hardware is stolen, you might use a full disk encryption solution like TrueCrypt or PGP WDE, or Ubuntu, Debian, or Fedora's built-in approach, and require a PIN or password on every boot.
If you just care about secure transmission, have code to ensure that you use transport security, and don't worry about encrypting the data in your database.