A password manager is a software utility used to store and manage the passwords you use to log into online services, such as your banking account. A password manager encrypts login information and requires a master password to decrypt that information so that it may be viewed.
Using a password manager is more secure than keeping a record of your passwords in a text file, or writing them on paper where they could be read by someone else.
Note that a password manager is still only as strong as the master password. If someone obtains your master password, they'll have access to all of your accounts. Protect your passwords carefully, and never give them to anyone else.
The image to the right shows an example of the KeePass login page.