To understand this page, you should firstly read : Bases of the genetics, The synthesis of proteins, The mitosis.

We saw that the DNA codes the synthesis of proteins. However these proteins, once produced in our cells, will act in our body. Our genetic inheritance thus determines the whole (called phenotype) of our physical and physiological natures.
The chromosomes going per pair, we have two specimens of each gene, and thus two alleles. These alleles can be, according to what our parents yielded to us, identical or different. One speaks then respectively about homozygote or heterozygote individuals.
Moreover, there exists for each gene a hierarchy of the alleles. I.e. certain alleles will take precedence over others at the time of the expression. When an allele takes precedence over another, it is called allele dominating, the other allele is called recessive. It also happens that two alleles can coexist without hierarchy, they are then codominating. For better understanding, let us take the example of the blood groups.
The blood group corresponds to the nature of one of the molecules of haemoglobin. One knows three alleles of gene corresponding to this molecule (and thus three different molecules). Allele A codes for an enzyme A which is functional, effective. The allele B codes for the also effective enzyme B. The allele O codes for the ineffective enzyme O on the other hand. With and B are codominants and are dominant compared to O.
There are six cases :

Alleles of the individualRelation between the allelesConsequencesBlood group
A and AidenticalThe individual has only molecules AA
B and BidenticalThe individual has only molecules BB
O and OidenticalThe individual has neither molecule A, nor BO
A and OA is dominatingThe individual has thanks to allele A molecules A. The presence of an allele O does not have any consequence, because the other allele is expressed.A
B and Oidem with B
A and BcodominatingThe two genes being with equality, they are expressed both. The individual has molecules A and molecules B.AB

So what happens when our DNA is modified ?
It should already be noticed that the genetic code (see here) is redundant : there are 64 different codons and 20 amino acids. Thus, certain amino acids are coded by several code different. Thus, the change in the DNA of codon CTT in CTC causes in ARNm the replacement of GAA by GAG. However these two codons correspond to the glutamic acid. This change thus does not have any influence on protein coded by gene. It is known as quiet change and does not have any influence on the individual.
In the same way, it can happen that a change modifies an amino acid in a protein. But if this amino acid is not located in the active zone of protein, it does not have an influence on the operation of protein*.
However, these changes are fewer. The majority of the changes involve the synthesis by the cell concerned of a protein either inactive or active but having an effect very other that the normal effect. Thus, one can have various consequences on the individual. Certain changes will have benign consequences, like albinism. Of other can involve more or less serious diseases like the drepanocytose : this disease causes deformations of red blood corpuscles and can involve decelerations or blockings of blood circulation, even of the infarctions. Lastly, certain changes will make that the cell behaves abnormally : it will be put to duplicate themselves without least control and to form a tumour then a cancer.

The modifications of the DNA can come from various sources. The majority are genetic : certain diseases are transmitted by the parents. Other genetic anomalies can appear at the time of the reproduction because of a change of the gametes. Certain changes are also caused by the environment. They are of two types. Some are caused by a dangerous product, as the tobacco which involves lung cancers or throat. Others can be caused by particularly intense radiations. Certain carrier waves energy can cause chemical reactions on the DNA, or damage it. One can quote the gamma rays (produced by the radioactivity) or UV, in particular in Patagonie, area located under the hole of the layer of ozone and where the number of cancers of the skin increased by 77 % in 15 years.

* The theory saying that the redundacy of the genetic code allows the quiet changes no to have any influence on the proteins has been questionned by an equip of the National Cancer Institut (United States of America). The researchers think that those changes don't modify the sequence of amino acids, but they can change the speed of the construction of the proteins, and by this way their shape (according to Science et Vie 1074, March 2007).