This is a list with several free password managers that you can trust. I will probably expand this list as soon as I discover other alternatives. I currently recommend you to use two password managers: KeePass and Password Safe.
Important: This article features only free, open-source software. It does not include paid alternatives. Make sure that you download these password managers from official websites. I guarantee you that they are clean, tested and recognized as pioneers in password management area. The best example: Bruce Schneier (a famous computer security cryptographer) started Password Safe project. I also ranked them by popularity, cross-platform versions and their history (see last table).
1. KeePass - most popular open-source password manager
Dominik Reichl launched KeePass project in 2003. He is the maintainer and developer of KeePass, one of the first tools from this niche. It has an extensive list of features such as: database encryption using encryption algorithms (AES or Twofish), browser support (it fills username and passwords automatically), built-in password generator etc. It would be worth mentioning that KeePass databases can also be locked using one-time passwords. 'OtpKeyProv' plugin (available from the KeePass plugins page) protects databases in such a way that one-time passwords need to be entered in order to open them. The plugin supports all generator tokens following the OATH HOTP standard (RFC 4226). Especially, YubiKeys support this mode (and I actually did test the plugin with a YubiKey, among others). You can also use a large collection of plug-ins and extensions made by independent authors. You can check the entire list here. KeePass comes in two versions: Classic and Professional. You can read about the difference between them on "Edition Comparison" page. There is also a large list of contributed (unofficial) KeePass packages for Linux, Mac OS X, PortableApps Suite, U3 Devices, Parted Magic, Spoon and even Preinstalled Environments (PE) here.
2. Password Safe - first open-source password manager
Password Safe is a project started by Bruce Schneier. However, since 2002 Rony Shapiro is the project developer and manager. This software benefits from almost all features listed at KeePass. Encryption schemes, a password generator, auto-filling of passwords in browsers and other features are available. Although the most popular version of this program is for Microsoft Windows, you can use a beta version for several Linux distributions. Password Safe also work with YubiKey USB-key but this is another subject. YubiKey is a product of Yubico company that incorporates Password Safe inside their USB flash drives.
How I ranked them
Both programs provide solid password protection. They are well-known, reputable trusted applications, using one or another is a matter of personal preferences. I have made this table with the main difference between them and explained why KeePass is on the first place and Password Safe on the second. I considered the following factors: project age, cross-platform versions, development rate, features, plug-ins, documentation, support, forum support and popularity.