If I make the SALT random for each user, how do I authenticate them?

It doesn't defeat the purpose of the unique salt to store it. The point of a unique salt is to protect your entire users repository from attack, not a given individual user. If an attacker compromises your database and is determined enough to crack a particular user's account, they will. There's nothing we can do about this. But they would have to spend an inordinate amount of computer time doing so - enough that it would not be feasible to spend that much time on each user - thus protecting all your users. Contrast this with using the same salt for all users - once the attacker has the salt, the same tables/processes can be re-run against every user in a relatively short time.

Salt is randomly generated for each user but it's saved somewhere in the database. You look up the salt for the particular user and use it to authenticate the user.

The point is, since salt is different for each user, you cannot use a prebuilt dictionary of hashes to map the hashed passwords to clear text (rainbow attack).

The salt prevents someone from getting a copy of your encrypted password database and mounting an offline attack against all of the passwords at the same time. It doesn't prevent attacks against a single password.

You might enjoy reading the original Unix password security article. It does a very good job explaining what a salt is, and why we have them: http://portal.acm.org/citation.cfm?id=359172