Using a sock puppet
You are using a sock puppet if you are secretly using several account on the wiki, reserving some accounts for certain types of posts, usually concurrently. The motivation is usually to protect the reputation of one account (the main) while being able to do less reputable posts with the other (the sockpuppet). A variant of this is to use a new account to see how people react to you if they don't know your history.
Using accounts in this way has often proved detrimental to the wiki and to the reputation of the puppet master, because a user's reputation is his or her stake in the community. It is no coincidence that the users who have been here the longest are usually the most reputable to the wiki. If the sock is used concurrently with the main, there are many opportunities to give yourself away, and strong interests to find out who you are, so often the sock is connected to the main eventually, and the damage to the reputation is greater than it would be had you posted with the main from the start.
If you feel you are limited by the role your existing account puts you in, try reshaping the role in other ways. Some editors have managed to change their roles like this, and even though it takes more time and is less absolute (some users will always identify you with your old role), the effect is more lasting and may in itself gain you respect with the community.
Not using a sock puppet
Some uses of alternate accounts are legitimate. For example, users who have tools to semi-automatically edit the wiki usually make accounts for them. The user page of such a "shoepuppet" identifies the main account it is connected to.
Posting as an anon is sometimes done because people can't or won't login, for example when using somebody else's computer. Sometimes the connection is disclosed on the IP's userpage, sometimes not.
Some accounts get abandoned. The password may have gotten lost and can't be recovered because there is no email address set up, or the address that is set up can no longer be accessed (change it in your preferences if you can), or a user simply wishes to make a new start, for whatever reason. While the endeavor is not always successful, the community usually tolerates this. This does not cover creating a new account to evade a ban on your old one.
Discovering a sock puppet
In the history of the community, accusations of sock puppetry have consistently been the result and/or precursor of much bigger drama than the sock puppets themselves have caused.
Thus, a public accusation of sock puppetry is a net benefit to the wiki only if the sock puppet is being used to damage the wiki or its community significantly, and you have certain proof.
If you do not have certain proof or are unsure how certain the evidence that you do have really is, please contact an admin privately with your suspicion (see GuildWiki:Administrator information); often, additional evidence can be procured.
Please refrain from asking new users whether they are a sock puppet of an established user. If they are not, this needlessly singles them out and confronts them with a downside of wiki editing before they have had a chance to feel at home in the community. Instead, consider privately asking the established user whom you suspect of controlling the sock.
Admins will not ban an account just because it is a sock puppet. A ban will be issued when the actual usage/behavior of the account warrants it (eg. vote inflation, causing disruption on the wiki etc). Depending on cirumstances, such a ban may extend to the "master" account, or it may not.