Most people have access to more than 1 computer (e.g. one at work/school and one at home). If a person logs on from work/school and then later from home, they will be using 2 different IP addresses. Some people may also visit friends and their houses. Some people may also go to libraries and internet cafes and various other places to access more IPs. Some people who are involved in the information technology field may have access to even more IP ranges. On top of all of that, many ISPs give access to a wide range of IP addresses.
All of this means that most people, without doing anything unusual, will have access to many different IP addresses. People are not pretending to be 2 different people in doing this - they are simply logging on from different places.
On Wikipedia, anyone can at any time create as many accounts as they like. A second account is called a "sock puppet". It is actually legitimate to create multiple accounts - provided that they aren't used abusively. Initially, anyone could have as many accounts as they wanted to, but later on various restrictions were put in. Essentially, using 2 accounts to gain more power than you should have (e.g. with voting) is the major form of abuse. Another thing that is against their rules is to harass someone. Lastly, and perhaps most importantly, creating a 2nd account when you were banned with your 1st account is considered to be abusive.
Wikipedia claims that they do not vote on anything, hence if that was true then sock puppetry would be irrelevant. However, whilst in theory there are no votes, in reality there are. People can pretend that they have more support than they do, and others feel compelled to support the majority. Indeed, if you stand up against a majority, you can find yourself being punished, even if you are right. Sock puppet abuse is a major way to game the system.
Essentially, therefore, sock puppetry is only really relevant because Wikipedia is used like a game. If Wikipedia was a serious project, sock puppetry would be irrelevant.
When someone is banned (or blocked - blocked in Wikipedia terms means that it is temporary), it is very common for administrators to make up stories that suggest that that person was using several new accounts, even though they probably weren't. No evidence is required - just a hunch is good enough. Indeed, they can just do it and they are almost never questioned about it.
If someone who was banned had logged on from work/school and from home, then that in itself is considered by Wikipedia to be sock puppet abuse, even though it obviously isn't.
Sock puppet accusations are also regularly used in AFDs - typically to justify deleting an article that really should not be deleted. Sometimes the false accusations work the other way though.
Sock puppetry abuse, but more importantly the false accusations of it, is one of the major problems with Wikipedia. It is far worse than multiplaying is on online games. It is insidious and ends up being much like a witch hunt.
It is made worse by the fact that most administrators of Wikipedia can't even see the IP addresses of people who use Wikipedia (unless they don't login), and to prove sock puppetry they must use Check User. Check User results are almost always secret, however, and their results can never be made public even when demanded of by the person who is falsely accused of sock puppetry. Nonetheless, Wikipedia people consider this to be "proof".