Alright, I'll explain this to my best.
We've all heard of DNA being referred to as a 'code' of sorts, but it's really more of a factory floor.
Specifically, DNA is 2 long chain of compounds (adenine, cytosine, guanine, and thymine) connected by 'strings' of sugars and phosphates, joined across the center. The two chains are essentially opposites; adenine on one chain binds to thymine on the other, and the same behaviour between cytosine and guanine.
Each of these compounds is chemically different from one another, and certain compounds will bind to some and not others.
DNA's purpose is simple- to produce proteins. Specifically, the two strands of DNA separate from one another, and a strand of RNA is formed in the rift- RNA is basically similar to DNA, but only with one strand.
The sequence of the compounds I mentioned determines the sequence of compounds on the resulting RNA strand. Amino acids then bind to the active sites on each of those compounds, forming proteins. The type of protein formed is determined by the specific sequence of compounds on the RNA strand, and therefore by the initial sequence in the DNA.
But this process isn't exactly perfect. Numerous things (radiation, presence of other chemicals, even nonstandard temperatures) can alter the process at any step- the actual production of protein can be affected, the transcription of RNA can be affected, or the DNA itself can be altered. It's a chemical reaction, similar to how acids appear to 'eat' metals- the compounds involved are switched out for different chemicals that are also present.
When this occurs when the protein is formed, or the RNA is transcribed, little difference is notable, as the effect is limited to a single strand of protein or a handful of strands.
When this occurs to the DNA itself, it is a mutation, and effects any subsequent protein produced by that strand of DNA.
Just about ANYTHING can be effected by mutation, as the proteins created by DNA form everything our cells are made of, and the behaviour of those cells determine everything- growth rate, hormone production, eye color, you name it.
But this in and of itself does not cause an entire organism to mutate- we're talking about a single strand of dna among many in each individual cell, although certainly the effects shouldn't be discounted altogether- for one, cancer is caused by an alteration of the cells reproductive mechanisms, causing it to divide rapidly, forming a tumor.
Whole organisms mutate generally when the mutation happens to (or is passed to) a sperm or egg, or the stem cells of a developing organism- cases where the altered DNA affects the whole body and not just an individual cell.
In these cases, there are essentially four outcomes.
1- Nothing or neutral. Some mutations have no effect whatsoever, or only change things from an aesthetic standpoint; If a red brick in a building is replaced with a yellow one, the only change is the color of that portion, it is functionally identical. It could even, in this example, just be a slightly different shade of red that isn't even distinguishable from the rest by the naked eye.
2- Beneficial*- The mutation conveys some form of benefit to the organism. Ie. The organism could form denser muscle tissue, giving increased strength and endurance compared to others.
3- Detrimental*- The mutation is somehow harmful to the organism, though not necessarily fatal. Maybe the lenses in the organisms eyes are mishapen ever so slightly and the organisms vision is blurred.
4- Fatal**- The mutation prevents or interferes with some vital function of the organism, causing it to die before it is able to reproduce.
The theory of evolution and natural selection is that an organism with a beneficial mutation will be better able to survive long enough to reproduce and pass its genes on. A detrimental mutation will lower the organisms chance to survive and reproduce succesfully. A fatal mutation will flat out prevent the genes from being passed along.
*- Note that beneficial and detrimental are relative cases, and in terms of evolution, are specific to ability to survive. As an example, the condition of sickle-cell anemia is a genetic trait that weakens the person afflicted. However, people with sickle-cell are resistant to malaria, as the parasite has difficulties surviving in sickle cells. Such things as coloration may not be seen as particularly beneficial or harmful, but in the case of the peppered moth, which has two subspecies, one dark in coloration and one lighter colored, the darker colored moths showed higher survival rates in polluted areas, if only due to being harder to spot by predatory birds.
**- It is also noteworthy that a number of genetic conditions can become fatal at some point, but the 'bar' as it were is survival to reproductive age and succesful reproduction. This can be seen in a number of organisms wherein reproduction itself is fatal to one or both of the parents- while this adaptation is fatal to the organism, they've already passed on their genes at that point, so it's not fatal in terms of reproduction.
Evolution itself is essentially an aggregation of traits beneficial to succesful reproduction that are passed on over time.