In Rails, one would generally use a pre-existing library. Authentication is easy to do wrong, and the problem's been solved so many times that it's rarely worth the effort to solve it again. If you are interested in writing your own implementation, then I'll describe how modern authentication works.
The naive method of authenticating a user is to store their password in a database and compare it to the password the user submits. This is simple but unbelievably insecure. Anyone who can read your database can view anyone's password. Even if you put in database access controls, you (and your users) are vulnerable to anyone who hacks around them.
Proper form is to use a cryptographic hash function to process the password when it is chosen and then every time it is submitted. A good hash function is practically irreversible -- you can't take a hash and turn it back into a password. So when the user logs in, you take the submitted password, hash it, and compare it to the hash in the database. This way, you never store the password itself. On the downside, if the user forgets their password, you have to reset it rather than send it to them.
Even this, however, is vulnerable to certain attacks. If an attacker gets hold of your password hashes, and knows how you hash your passwords, then he can make a dictionary attack: he simply takes every word in the dictionary and hashes that word, keeping it with the original. This data structure is called a rainbow table. Then, if any of the dictionary word hashes match a password hash, the attacker can conclude that the password is the dictionary word that hashes to that password. In short, an attacker who can read your database can still log in to accounts with weak passwords.
The solution is that before a password is hashed, it is combined (usually concatenated or xor'd) with a value called the salt which is unique to each user. It may be randomly generated, or it may be an account creation timestamp or some such. Then, an attacker cannot use a rainbow table because every password is essentially hashed slightly differently; he would have to create a separate rainbow table for every single distinct salt (practically for each account), which would be prohibitively computationally expensive.
I will echo the advice of the other answerers: this is not simple stuff, and you don't need to do it because it's been done before, and if you do it yourself you stand a very good chance of making a mistake and inadvertently compromising your system's security. But if, for whatever reason, you really, really want to write one yourself, I hope that I have provided an (incomplete!) outline of how it's done.