Pseudonymity. A pseudonym is a self-designated, unauthenticated (fictitious) identity. It may or may not be traceable to a real person (Eg: Paul French, Mark Twain, George Sand, Richard Bachman), or group of persons (the academic shared pseudonym "Nicolai" used in the mathematics community in the 19th century, Aesop, Frank Dixon, Alan Smithee). Traceability is not a crucial characteristic, though it's an important one. Pseudonyms can be plucked from the air, and dropped once their usefulness has been spent. The key concept is that a pseudonym is a self-selected identity, not bestowed or authenticated by an external CertificateAuthority.
Serial pseudonymity. Pseudonyms are often used more than once, like a PenName. Each serial use of the pseudonym builds a reputation attached to that pseudonym. Sometimes the explicit intent is to create a reputation, a DramaticIdentity, divorced from the holders' personal reputations. In other cases, the serialization is inadvertent. For instance, criminals that have signature modus operandi can gain pseudonymous reputations--e.g. Jack the Ripper.
There may be no guarantee that a SerialPseudonym? actually corresponds to the same person(s). However, the CypherPunks are concerned with making this guarantee cryptographically secure. (See PublicKeyInfrastructure.)
Anonymity in this usage refers to communication that is not attached to any real person or persistent identity. For example, this unmarked contribution to MeatballWiki will become effectively anonymous once the diff is lost, as it becomes impossible to determine who wrote what. It may even have several anonymous authors.
Anonymity can be considered a special case of pseudonymity. There is virtually no difference between a (non-traceable) one-time identity, and pseudonymity. If the only reference to an agent is a unique confluence of events, then it may as well be anonymous, provided none of the components of the identity may be traced to the source agent.
See also UseRealNames, UseRealNamesDiscussion, the ImportanceOfIdentityInOnlineCommunities.
I have long believed that people create pseudonyms partly because they discover that they aren't the only "John Smith" online. Thus, in order to differentiate oneself from the masses, one needs a unique "handle." Personally, I don't feel this need because there are only a few dozen (at most) Sunirs in the world.
Another possible reason is that people want to have a DramaticIdentity. Well, an identity that is dramatic. I have more recently come to believe that the RealWorld is awfully dramatic if you make it so. So, once again, I don't need a handle. -- SunirShah
There's been some discussion in the popular media of "the brand of self" -- promoting yourself as a distinct brand. To wit, having a distinctive name is a helpful handle -- it turns out that my combination of relatively rare German and Welsh names makes for a rather handy tag on this newfangled Internet thing (OK, raise you hand if you found this page by searching for my name). Though WikiNotation? does tend to destroy this somewhat by RunningWordsTogether?. -- KarstenSelf