Usually, people think of authentication in terms of passwords. Although passwords are frequently used for authentication, there are actually a variety of authentication mechanisms. These mechanisms can generally be categorized as verifying one or more of the following:
This is the field of biometrics, including techniques such as fingerprint scans, retina scans, voiceprint analysis, and so on.
This is a traditional password system.
This includes mechanisms such as challenge-response lists, one-time pads, smart cards, and so on.
Some systems combine these approaches. For example, a smart card that requires the user to enter a personal identification number (PIN) to unlock it is a combination of something you have (the card) and something you know (the PIN). In theory, it is considered a good idea to combine at least two mechanisms, because people can steal either one: the thing you have is susceptible to ordinary theft, and the thing you know is compromised by sniffing if it passes over the Internet; but it's rare for somebody to be able to get both at once. Automatic teller machines use this combination; however, ATMs also demonstrate the flaw in the theory: when you are authenticating (standing at the ATM), you reveal what you have (your card) and what you know (your PIN) simultaneously, making yourself vulnerable to a thief who watches you use the machine to capture your PIN, then steals your card as you leave.
There are many types of biometric systems in use or under development today; they test such diverse personal characteristics as your voice, your fingerprint or handprint, your retina, your signature, and your typing patterns. Biometric systems are extremely attractive, because they get around the problems associated with using things that can be stolen or revealed. (Even the horror movie scenario of hacking off somebody's thumb to use in the fingerprint scanner is taken into account; most scanners insist that a pulse be present.) Unfortunately, biometric systems are not practical for normal Internet applications.
Commonly available computer hardware is not capable of reading fingerprints, much less retinas. Voiceprint technology is more tempting; it's not unusual for machines these days to have microphones. But it's not universal, either, and you can't guarantee that every machine you'll want to log in from has a microphone, much less the client application to digitize and transmit your voice. Moreover, every machine may not have the disk space and network bandwidth needed for this approach. Finally, if the biometric information has to be communicated from where you are to where it can be checked, you run the risk of an attacker's capturing it and replaying it later, unless you have some way of encrypting or otherwise protecting it.
Even if every machine had all the capabilities it needed, reliable voiceprint identification is surprisingly hard to accomplish. Many people wish to use their computers even when they have head colds, which throw off many voice-recognition systems. You can't use a fixed phrase as a password, or you would be vulnerable to a literal playback attack involving a tape recorder. (These problems have been addressed in dedicated systems, but there are no widely available authentication systems for general-purpose computers that can deal with them, and it's not at all trivial to develop one.) You can't use a different phrase every time, because voiceprint comparison on arbitrary text is not a real-time operation - unless you have a spare supercomputer lying around to do it with. Almost everybody wants to log in with a delay measured in seconds, at worst; getting back to the user with an answer in a few hours is not going to work.
Keystroke timing, used to characterize someone's typing pattern, is a surprisingly reliable biometric identification system and requires no special hardware. The computer provides a phrase to type, and then times the gaps between characters as you type it. On a phrase of reasonable length, people type in an identifiable pattern, and imitating it is much harder than it looks. Like voiceprints, keystroke timings may change for environmental reasons; in the case of keystroke timings, colds aren't a problem, but people have great difficulty authenticating while drunk. (This is not necessarily a disadvantage, of course.) The genuine disadvantage is that keystroke timings can be gathered only at the machine to which the keyboard is attached. That means that use of this method across the Internet requires modified clients. In addition, there are some people, mostly poor typists, who have trouble ever authenticating because their typing patterns are inconsistent.
If the long-lost heir to a fortune turns up in a novel, you can bet that fingerprints aren't going to be available to do biometric authentication. Instead, the claimant is probably going to try to prove that she is who she says she is because she knows the name of the stuffed animal she slept with at age three. This is authentication by knowledge, just like the traditional UNIX password system. (And, just like traditional UNIX passwords, the claimant's answers can be faked if she's a good guesser and gets enough tries.)
Authentication that depends on something you know relies on that something's being both hard to guess and secret. In order for you to authenticate reliably, you have to know the secret reliably, too. This isn't as easy as it sounds. Most people are bad at making up and remembering unguessable things, and they're worse at keeping secrets. If you use short keys, it's easy to guess them; if you use long keys, it's hard to remember them. If you write them down, you're basically converting to a different type of authentication; now, it's something you have.
System administrators who unblushingly tell their users never to write down passwords probably have a few stashed in their wallets anyway; this is a combination of "what you know" and "what you have." "What you know" is how to read your own handwriting, and which slip of paper contains the passwords, rather than last week's lunch order. "What you have" is the slip of paper itself.
Despite all of the risks of "what you know" systems, it is still practical to use such systems, as long as you aren't revealing the secret to everybody in the near vicinity every time you authenticate. There is a reason why passwords are so popular: they're fast, they're cheap, and, in practice, people don't forget them or lose the pieces of paper all that often. However, it is absolutely impractical to pass them across the Internet in any form that can be used safely.
Is there any way to use a "what you know" system safely on the Internet? Yes. Use passwords, but make sure that they are nonreusable (one-time) passwords.
There are two ways to make traditional, memorized passwords nonreusable. One is to include an encrypted time-stamp - this is the method that Kerberos uses. As long as you can't modify the time-stamp without knowing the password, this prevents playback. Unfortunately, it requires two things:
Special client software that knows how to time-stamp the password.
Synchronized time between the client and the server. If the server and the client have different times, the password will be either invalid already or not yet valid when it arrives.
The other way to make traditional, memorized passwords nonreusable is to use a challenge-response system. With such a system, the password you give depends on a prompt that the server gives you. Challenge-response is the traditional method for identifying yourself at a speakeasy, entering a military camp, or meeting fellow spies in art museums. It may seem to be an impractical approach, because in order to be reasonably safe from playback attacks you need a wide variety of challenges, each with a separate response. If you have trouble remembering one password, you certainly aren't going to remember 47 of them.
In fact, however, challenge-response systems for computers are designed so that instead of memorizing the response to each challenge, you memorize a rule for converting the challenge into a response. This concept has yet to become popular as a pure "what you know" option, because it's difficult to apply to a large number of people. A rule like "reverse the first three letters, capitalize the fourth, delete the fifth, and uppercase the rest of them" is easy to program, but it's not necessarily easy to represent so that you can have a different rule for each user. Even if you can remember your own rule, you're apt to follow along the challenge with a finger, muttering the rule to yourself while you try to implement it; someone watching over your shoulder can pretty easily determine the rule.
In practice, the most successful authentication systems for Internet use today are based on the third type of authentication: "something you have." What you have may be a printed list of single-use passwords or an electronic card; it's usually, but not always, combined with "something you know."
The use of electronic cards makes it practical to use challenge-response or time-based encryption. For challenge-response, you encode in the card itself the rule that converts the challenge to the response. For time-based encryption, you encode in the card both the time source and the encryption smarts. Either way, you can do authentication without needing modified clients or users with trick memories.
Using printed lists of passwords is generally referred to as using one-time passwords. This confuses people because all nonreusable passwords are good only one time, by definition. Printed password lists are called that because they resemble a spy device called a one-time pad that consists of a pad of paper with different instructions on each page. You encrypt one message with each page, so that the opposite side doesn't get multiple chances to break the code.
 In fact, printed one-time password lists resemble one-time pads so closely that it's inadvisable to bring one to the attention of a customs agent when travelling internationally.