Passwords are backed up to the cloud, but in an encrypted way. Therefore your passwords will sync across all or your devices (which I agree is essential convenience). However, your master password never leaves your computer. A good litmus test is that if you loose your master password, there is no way for LastPass to reset it. They may have a less secure option in which they can reset your password (and hence have access).
My only gripe with LastPass is that it's not fully open source, which is a BIG GRIPE when it comes to security critical software. But I'm not aware of an open source option that provides syncing across devices.