|I use the free version of LastPass, which is a browser extension available for almost every browser. The passwords are stored in an encrypted vault locally; the first time you use the extension, you create an account on lastpass.com and a strong master password for the local vault; encryption and decryption keys for the vault are then created based on that master password. |
I saw the comment above about the dangers of having an account hacked, but I don't think that's an issue with even the free password managers as long as they use a strong encryption algorithm on the data before sending it to the cloud.
With LastPass (and possibly others), the master password and encryption keys remain on the local device, and the "vault" (really just a blob of encrypted data) is backed up to the cloud so that you can use it with other devices. Of course, you would need to use the same master password when setting it up on the other devices, because that is what the encryption/decryption keys are based on.
The fact that your master password and encryption/decryption keys are only stored locally is a double-edged sword. On the plus side, it's extremely secure - even if hackers manage to get into your lastpass.com account, all they will be able to get is an encrypted blob of data with no way to decrypt it.
On the other hand, if you forget your master password, you're basically hosed because all lastpass.com has is an encrypted blob of data with no way to decrypt it. Their premium (pay) version includes something called "Emergency Access", which I'm guessing is a way for you to recover your master password on your local device. The only thing the free version offers is an ability to add a reminder that will hopefully help jog your memory; if it doesn't, bye-bye passwords.
I'm really happy with the way it works - it logs me into my frequently used websites with a single click, without me having to remember the URL or type in a username and password. I'd recommend it highly, with the caveat that you use a strong master password - such as an obscure phrase or an abbreviation of a phrase - that you can remember.