Recs.
Updated
KeePass is a free cross-platform password manager available for Windows, Linux, and macOS. There are unofficial ports for other operating systems and mobile devices.
SpecsUpdate
Pros
Pro Extensive plugin and extension support
Keepass offers a range of plugins and extensions in different categories: Integration & Transfer, Automation & Scripting, Cryptography & Key Providers, and others.
Pro Customizable password records
Users may define any extra fields (eg. "Surname", "Credit Card No", "Start Date", "Expiry Date" and "Security No") and their values may then be easily copy-and-pasted when required. This is a great feature not found in a lot of password-managers: they leave you to consign such additional information to the "Notes" section.
Pro Powerful built-in synchronization mechanism
KeePass v2 has a built in sync mechanism that allows setting appropriate level of sync, kind of sync, and choosing between a number of sync sources (Dropbox, Drive, etc).
Pro Works inside Android apps
There are variations in support of features, so this is dependent on the Android app being used. Keepass2Android is one that has built-in synchronization over ssh and other protocols, fingerprint support and plugins are also supported to a certain extent.
Pro Offers multiple sync options
Since the database is stored locally it can be synced across devices using a cloud storage service like Dropbox, Google Drive or via a file sync tool like BitTorrent Sync. KeePass v2 also provides a powerful built-in synchronization mechanism that allows setting appropriate level of sync, kind of sync, and choosing between a number of sync sources (Dropbox, Drive, etc).
Pro Built-in history that tracks changes to individual credential entries
With the built-in history you can lookup previous passwords, track changes in general and if necessary recover those. In essence you are looking at a built-in Version Control System (VCS) which operates on single entries. Browsing the history of specific entries also displays associated metadata like the date of a change.
Pro Allows storage of attachments such as pictures, documents and all kinds of files in general
Appending attachments to individual credential entries allows things like scanning important documents (for instance a scanned document containing the login information for your internet connection), appending them to the corresponding credential entry and afterwards destroying the original paper copy.
Pro Synchronization across devices
Since the database is stored locally it can be synced across devices using a cloud storage service like Dropbox, Google Drive or via a file sync tool like BitTorrent Sync.
Cons
Con KeePass in OSX is not user friendly
There is no official installer for OSX and install steps are not well documented. Mono under OSX does not perform well. The keypass UI is also hard to read because it does not scale well in OSX under high resolutions. However, there are a couple alternatives: KeePassXC, which is a fork that's a native OSX application, and MacPass that provides a number of comparable features to the official KeePass client.
Con Officially supported only on Windows, Mono or Wine
While there are ports available for Linux, OSX, iOS, Windows Phone and Android, the only officially supported version of KeePass 2 is on Windows and all platforms running Mono like Linux, Mac OS X, BSD. KeePass 1 is supported on Windows and Wine.
Con Not all plugins are working on Linux
Not all plugins are working on Linux, so not all features are available on Linux (as tested with Keepass2).
Recommendations
Comments
Flagged Pros + Cons
Pro Free and open source
KeePass being open source means that a number of people have reviewed the code and found it to be secure.
Con Uses a database
Uses a database and thus decrypts all passwords within it when only one is needed. Which is less secure than having individual files. It also means you will have only one master password.