One common logical fallacy that people use in arguing (especially online) is using anecdotal evidence as actual evidence. This means that they find some example of something that corresponds with their argument, and use that example as evidence to support their argument.
You might think at first, "what's so bad about that? They're using evidence to support their argument, isn't that what you're supposed to do?" Yes, I suppose this type of argument is better than just pulling something out of your ass with absolutely no evidence behind it (which is probably just as common online as anecdotal evidence). However, that doesn't mean it is good. The issue is that for pretty much any side of an argument you can find an example that supports that argument. A common one that is loved by the Canadian Pirate Party is, "so-and-so artist says that he/she likes piracy because it boosts concert sales, therefore piracy is good." This could be a legitimate argument, and economically it is possible - reduction in the price of a good (the music) increases demand for a complement good (the concerts), but without detailed statistics the effect is ambiguous to the musician's overall welfare. The problem is that one example alone does not provide a justification for the argument. Instead you would need a lot of data from a good sample to verify the actual effect.
From a more formal standpoint, the root of the problem is a sampling bias. You are taking a potentially non-representative subset of the population and claiming that it is representative. There is no guarantee that this particular anecdote is representative, so any conclusion you derive from the anecdote about the population isn't really valid. In the example in the last paragraph, they're using a subset of a subset of the population (a few musicians do not represent all artists).
This isn't to say anecdotes are all bad. You can derive useful insights about larger phenomena by analyzing anecdotes, for example asking "why do certain artists like piracy" can lead you to the basic economic analysis that I mentioned earlier. However it ends at the insight, if you want to derive a conclusion about the population with this insight, you have to resort to more advanced statistical methods.