Someone else briefly mentioned this, but I'll state it more clearly.
There are a lot of things that the human body is wired to release endorphins (i could be wrong on the specific chemical, but it's either endorphins, dopamine, or something) to reward doing certain things (e.g. eating) so people will do it, and not die. There are things that aren't actually positive to a human that are rewarded anyway, due to glitches and things. One of these is self-mutilation. Drugs are another. In fact, any positive experience (or at least, any experience that makes you feel better) will release this chemical.
Now, some people with certain mental and biological factors (I don't think we know which is more powerful, nature vs nurture, etc) are predisposed to become addicted to this chemical. If they do any one thing enough times that they realize they can get it from it (e.g. drugs, eating, or yes, even video games) they may become addicted. Drugs and cigarettes will be more addictive, as they also have other chemicals in them, that achieve the same purpose.
So, one can be addicted to pretty much anything they take pleasure in. Very few people (relatively, at least) are actually addicted. The difference between being addicted to something, and merely liking something, as some of you have been arguing about, is whether you are seeking the effect of the action, or the action itself. For example, I have moderate allergies. I take a Zyrtec every day. I cannot function very well without one. Am I addicted to Zyrtec? No, because I am trying to alleviate my symptoms, I'm not just trying to get a Zyrtec in me. With an addiction, it's different. Let's take the classic video game example. Say I enjoy playing World of Warcraft. This may be because I'm just bored a lot, or maybe I am one of the less lucky among us, who doesn't get as much social interaction as I would like (which there is definitely no shame in). Either way, let's say I play WoW for a few hours a day. This is not addiction. However, as soon as I cross the threshold from 'I don't want to be bored for the next few hours' or 'I want to talk to my guildies' to 'I want to play World of Warcraft', I am addicted.
Wow, this was longer than I thought it would be. Either way, I learned all of this from Stuff You Should Know, so if it's wrong, blame Josh and Chuck, not me.