A password manager is a software program which encrypts and stores various passwords for a user. The user has a master password which unlocks the "keyring" and enables the decryption and use of any stored password. When a user visits a website which requests a password, the password manager will auto-fill the password for them.
Password managers allow and encourage users to choose strong passwords with long sequences of random characters. Some password managers include a strong password generator as well. Additionally, typing uppercase letters and symbols on mobile devices can be tedious. Having software that auto-fills these characters is helpful.
The one drawback to using a password manager is that it could be vulnerable and would yield many passwords as a result. The master password becomes a valuable target. The password storage could be compromised or decrypted or the password manager software itself could have a vulnerability. However, most of these vulnerabilities would require direct access to the computer, making it a small risk when weighed against the security advantages.
Wikipedia has a list of password managers. The most frequently recommended are LastPass, Dashlane, KeePass, and 1Password.