Explain how mutations can become established in a gene pool? Why does the frequency of some mutated alleles increase while others decrease in frequency over time in a gene pool of a species?

A mutation can arise from a change in the DNA sequence. If a mutation occurs in a germ cell, the mutation can be passed along to the next generation.
Some mutations are harmful, some have no effect while a small percentage may actually prove beneficial.
In the Theory of Natural Selection put forth by Charles Darwin, it is mentioned that organisms possess variations. These are due to gene mutations. If an organism has a variation that makes it better suited to the environment in which it lives, it may win the struggle for existence, survive and reproduce. Possibly, its variation which is an adaptation to the environment may be passed down to its offspring. In this way, the frequency of the gene that produces this variation may increase over time. However, if a gene mutation is not beneficial, it may become less common in the gene pool because it doesn't benefit the organism or improve its chances of survival.
An example demonstrating how gene frequencies can shift can be seen in the Peppered Moths in England. Before the Industrial Revolution, most moths in the area where the study occurred were of the light-colored variety. However, a few dark-colored moths existed due to a gene variation. Light ones blended into the light- colored background of the trees upon which they rested, while dark ones stood out and were eaten by predatory birds. In this way, natural selection caused the gene that produced the light-colored moths to increase in frequency, while the gene producing the darker-colored moths decreased.
During the time of the Industrial Revolution, the trees in the forest became covered in soot. Dark-colored moths now had an advantage because they blended into the background and escaped predatory birds. These moths lived and reproduced and the gene frequency increased for the dark variety. However, the lighter moths no longer had an advantage and were consumed more frequently. Their numbers due to the selective pressure of predators were drastically reduced.
Life doesn't operate in a vacuum. Because the environment changes, gene frequencies can change over time due to natural selection. I have included a link which elaborates the story of the peppered moths. I have included a second link summarizing Darwin's Theory of Evolution.
 
http://www.millerandlevine.com/km/evol/Moths/moths.html

http://www.pbs.org/wgbh/evolution/educators/course/session2/explain_c_pop2.html