KeePass
An upvote for KeePass that I've used for years - open source, uses a local encrypted file and not tied to any service. To unlock the vault can just be a master password, but it can also require access to a specified file, perhaps on a remote USB stick for extra security.
It runs on on all desktop OSs and there's an app for Android, not sure about iOS. On Android it can auto fill if required and makes it easy to copy and paste if not.
I chose to share my password file on Dropbox to keep the versions on the phone and desktop in sync - yes, that's a security hole I've opened but my choice to balance convenience. I'd use OwnCloud if I had it set up.