Your first paragraph is correct. Your second one borders on nonsense.
There is already a thread on evolution, which is probably where this question should have been posted (some of the answers may already be there). But I will answer your question about what a mutation is here.
At its broadest definition, a mutation is any change in the DNA. It can take several forms: a copying error, that changes the sequence but does not change the number of base pairs. A deletion, which removes one or more base pairs from the sequence (and may shift the reading frame). An insertion of one of more base pairs into the sequence (which may also shift the reading frame). Fragments of genes, whole genes, parts of chromosomes, and entire chromosomes sometimes get duplicated accidentally- but mostly, when you hear the word "mutation", people are usually talking about it in its more restricted sense of changes in the base pair sequence of a single gene.
The evidence for mutations is overwhelming. Mutations happen all the time. Roughly one in a million base pairs is copied incorrectly every time a cell divides. You're probably a mutant. I'm probably a mutant. But those mutations make up a very small part of our overall DNA.
There are two basic effects a mutation can have: it can either change the amino acid sequence of the protein it codes for (thus possibly changing the way the protein works for better or worse), or it can disrupt the gene to the point where the protein it codes for can no longer be produced. Often these errors are lethal in embryo or in utero, so these mutations never get passed on. Some are neutral, and a very small amount are advantageous.
A classic example of a single point mutation is the change from GAG to GUG in the third codon of the alpha hemoglobin gene. GAG codes for the amino acid valine at the 6th position in the protein specified by this gene but the GUG mutation codes for glutamic acid instead. Changing this one base pair (A->U) changes the codon (GAG->GUG) which changes the amino acid it codes for (Valine->Glutamic acid), which alters the structure of the resulting protein chain enough so that it misfolds badly and can't carry oxygen efficiently, and the result is sickle-cell anemia.
That's mutation. Selection and fitness are entirely different topics.
EDIT: More precisely, the mishapen hemoglobin exerts its major effects by causing a change to the shape of red blood cells carrying it, so that they taken on a crescent ('sickle') shape instead of their usual round donut shape, and can cause logjams in narrow capillaries. The lowered capacity for oxygen transport is not the primary problem.