LastPass is clearly the best app for password management, and for good reason.
One of the first well-rounded password managers available, and one of the first that really made it easy to store all of your passwords either online and synced with other computers and devices, or locally on one device.
In short, LastPass remembers your passwords so you don't have to, and makes it easy to audit your passwords, use stronger passwords in general, and even automatically change a password for you if a service has been hacked or compromised.
LastPass supports two-factor authentication for your password vault using Google Authenticator, USB devices (using a method we've outlined before), or a YubiKey, The service picked up a much-needed update a year or so ago to streamline the UI and make it easier to use, and sports a number of additional features like credit monitoring, secure password and document storage (and sharing), notifications when a site you have an account with has been hacked, tools to autofill forms and streamline online shopping, and more.
LastPass supports Windows, OS X, Linux, Android, iOS, Windows Phone, and Blackberry, and has plugins for Chrome, Firefox, Safari, Opera, and Internet Explorer. It's free to download and use, but if you want its best features and the mobile apps, you'll need to upgrade to LastPass Premium, at $12/yr.