How can you extract individual characters of an encrypted string such as a password? Many banks and other systems have some sort of a  "select the Nth, Nth, and Nth characters of your password" system.

I don't believe they are going to be storing in plain text, nor would it make much sense to hash and salt each character of the password. Someone I asked mentioned they believe there are algorithms that can check or manipulate parts of encrypted data.

Do these algorithms exist and how do they work?
Most financial services companies that I've come across who make use of this kind of system are encrypting the password using a Hardware Security Module ("HSM").

In this kind of system the password is symetrically encrypted with a key held in the HSM, and then stored on disk.  The application will then pass the encrypted password and the individual characters into the HSM which will decrypt the password, make the comparison with the supplied characters and return a true/false answer to the application.

The security of this kind of system relies on the intended properties of the HSM which should be resistent to tampering and which run a very limited number of functions to reduce attack surface.

Also in this kind of situation key management is obviously very important as compromise of the key would allow for easy access to all customer passwords. This tends to lead to split authority and key parts being stored in safes, as a backup for the HSM.