Evolution is a concept that links genetics, history, and adaptive selection. All life evolved from common ancestors, the earth had been around for 4.6 billion years, simple cells (prokaryotes) have been around for most of this time for 4 billion years, and complex cells (eukaryotes) have been around for around 2 billion years. This journey is never ending, because even if all life on earth was wiped out today, the molecules that we are all made from would find a way to do it all again using evolution as a tool. Species are evolving constantly on a daily basis, however these changes are small and rarely observed because genetic mutations that occur are usually harmful to organisms, we only have to think of cancer cells as an example.
All life strives to survive, many scientists explain most animal behavior based on this genetic desire to out compete their rivals, and their rivals are all who do not share a large proportion of their personal DNA, therefore family members, especially offspring and siblings, who both share 50% of their DNA are usually cared for, and members of their own species are tolerated, and sometimes helped, if only for the chance that they might help you too in the future. Survival is worked towards, and all life has a niche within its environment which it exploits, competition occurs between different species occupying the same niche, usually one species is out competed, and this is where evolution can create advantages for particular species, in order to win, and survive.
Genetic mutations occur on many levels, during gamete production whole chromosomes can be lost or gained in genomes, such a Down syndrome where an extra part or whole chromosome 21 is present in the genome. Mutations can also occur through genetic cross over by chromosomes when the cells are dividing, this causes genetic information to be transferred to different areas of chromosomes. Point mutations occur within the genetic material itself, such as duplications, deletions, and insertions of the bases in the DNA, this is usually caused by miss-reading during transcription, and synthesis of DNA.
These genetic mutations are the core element in both rapid and slow evolution, mutations in organisms are mostly harmful, but under the right selection pressures, such as a forced change in diet or habitat, these mutations, such as slightly longer legs, or beaks are rapidly selected for and evolution suddenly appears to be no longer a slow process, and changes can be measured and viewed such as in studies on Darwin's Galapagos Finches.
Mutations are not the only driving force to change, but sexual selection plays a huge role in both maintaining the fittest genes within a species, and also selecting unusual traits which would help its offspring to survive. If we consider the peacocks tail, this is a purely sexual adaptation, as a tail so heavy and resource consuming has no positive influence on the survival of the individual peacock, and usually it would be selected against. Sexual selection obviously is playing a role here, female peacocks are choosing the males with the largest tails because it shows they can survive even with this burden, its like a test of peacock'-hood, and so their larger tail genes are passed on, and you can imagine, over time the tail feathers getting longer and longer through female selection.
Evolution is a process, it is influenced by various elements, but the fundamentals in evolution are that species are constantly evolving, and although mutations can be increased by mutagens, these mutations usually harm an organism. The underlying genetic units of change can not be slowed, as although the phenotypes of a shark have not changed for millions of years, it is still evolving and it's current essence is just being constantly selected for.