A mature open source project (GNU GPL version 2), KeePass is a free password management solution for Windows, OS X, or Linux, running natively on Windows and requiring Mono for the other platforms. Many of the benefits of open source software are prevalent in KeePass, including ports to other client operating systems and a robust plug-in ecosystem. With the extensibility offered by plug-ins for KeePass, you can change the encryption algorithm, automate logins through your browser, integrate an on-screen keyboard, or even create scripts you can run against KeePass.
KeePass was designed to store a local copy of the password vault. Cloud backup and support for synchronization across multiple devices are obtained through plug-ins that work with the likes of Dropbox, Google Docs, and Microsoft OneDrive. A side benefit of a local password database such as KeyPass is the ability for multiple users to share a database or for one user to keep multiple databases, sharing some and keeping others private.
Mobile support for KeePass is a little more obtuse than some of the commercial options. Ports are available for iOS, Android, and Windows Phone, but the big question becomes synchronization support. Not all mobile ports support cloud synchronization, and those that do support only a subset of the cloud options. Some mobile KeePass clients carry a cost, though most are in the $1 to $2 range.
If you're more concerned about the security of your password vault than mobile clients and device synchronization, you'll be pleased to know that KeePass supports multiple authentication methods by default. KeePass database files can be locked by a combination of password, key file, and Windows user account. With a key file stored on removable media such as a USB thumb drive, two-factor authentication can be used to secure access to your critical passwords.
The biggest downside to KeePass is complexity. Getting all of the advanced functionality offered by the competition will require quite a bit of research, setup, and maintenance. While KeePass is a great solution for fans of open source, maximum flexibility, and free software, it is certainly not as straightforward as some of the cloud-based services listed here.