A mutation is the change in the genetic code whether it be through the DNA or the RNA. In essence, the many base pairs continue to appear and appear over the chain and from time to time, there is an error. However, what exactly is a genetic mutation? Before I can answer that, I want to propose an idea to the reader that might provoke an uncomfortable thought, but consider it for a moment before completely shunning it.
In the comic book, X-Men, the mutants are shunned because they have these super powers and because there was a 'genetic' error that made them that way. The interesting thing, actually, is that we are all mutants in respect to the fact that genetic mutations happen all the time. Now, are they as bad as the ones in the comics? No. But, they do occur.
When proteins are made, the DNA transfers information to the mRNA which transfers information to the tRNA which transfers it to the rRNA which, in turn, makes the protein. Now, let's assume for arguments sake that we have a set of base pairs that looks like: ATCGGCAT. When it converts into RNA, it becomes UAGCCGUA. And then it converts into the next RNA and it becomes: AUCGGCAU. And it continues until you get to the protein.
A genetic mutation occurs when there is a sudden issue in the connecting of the base pairs. Perhaps one of the cytosines did not connect correctly to a guanine and voila, you have a genetic mutation. Because of this genetic mutation, whether it happens at the start of the process of creating new amino acids or at the end of the process, the end result is totally different.
Does this mean that we are going to all grow wings and fly out of our father's biology lab like Angel did in the X-Men movie? No. But, what it does mean is that my hair color will be different than yours. If there were no genetic mutations, we'd all look exactly the same. However, the reason that people fear 'genetic mutations' is because it could result in catastrophic situations. Sickle cell anemia was a genetic mutation. Now, it can be passed down from person to person and that can be a traumatizing thing. The same is for other diseases.
When you're dealing with so many millions of base pairs, it is accepted that there is going to be a mistake. If you were typing a five hundred page essay, there is obviously going to be a few errors in it. And, if you didn't have spell check, chances are, you'd never catch it until the reader sat down and said, "Wait, that's wrong." The same goes for the genetic code except, it creates a bit more changes in the body.