Basic question about evolution
I want to understand a mechanism how at the first times when the life began on earth evolution helped the new species to arise through selection while the same organism also continued to live as not evolved.


Well, this beginning is not grammatically correct so it is hard to make much comments on it.


  If natural selection favors survival of the fittest, [..]


Selection is the process by which fitter individuals increase in frequency in the population. survival of the fittest is a layman phrasing of the process that is natural selection. Natural selection therefore does not favour survival of the fittest, but it is represented by the phrasing survival of the fittest.


  [..] th[e]n why after a living evolves to a new fittest form, the older form still exists


It does not. The older form is called an ancestor. The ancestor does not exist today. If there was a speciation events, then the ancestral lineage would have giving rise to more than a single lineage today. Some of these modern lineages may 'look more' like the ancestors than others but it does not mean that any ancestor still exist.

For example, the chimpanzee is not an ancestor of humans. Chimpanzees (and bonobos) and humans share a common ancestor that is about 8 millions old. To consider another example, humans and an oak tree share a common ancestor. This ancestor was neither a human, nor an oak tree. This ancestor lived about 1.6 billions years ago.

If you want to know why I am using term 'lineage' rather than 'species', you can have a look at this post

See below linked Biology.SE posts for more information


  the older form still exists while it should be counted as unfit for survival and extinct


Don't forget that different lineages may (and will be at equilibrium due to the competitive exclusion principle) be adapted to different ecological niche. As such, one lineage will not necessarily outcompete the other one.

Source of information on StackExchange

There is a lot you can learn on the subject. You should start with an introduction to phylogenetics such as the one provided in the post


Why do the ancestors of birds still exist?


You can also have a look at 


Extinction of species


You should also have a look at 


How does Darwinian Evolution work?


Reading your post, it sounds like you may have too a pan-selectionist view of evolution. Related to this point, you might want to read


Why don't mammals have more than 4 limbs?


Source of information external to StackExchange

Understanding Evolution is a good introductory course to evolutionary biology.



Response to comment


  Thank you for the very descriptive answer, as you have figured out easily I am very new to evolution. I will read the references you have provided carefully


You're welcome!


  is after evolution of a specie through natural selection, 


This show again that you have too a pan-selectionist view of evolution. 


  unfit ancestor extinct and this is explained through phylogeny ?


From wikipedia


  Phylogenetics is the study of the evolutionary history and relationships among individuals or groups of organisms (e.g. species, or populations).


A phylogeny is a tree of life of a given taxon. For example here is a phylogeny of placental mammals.

To be able to fully answer your own question, you need to understand the relationships between lineages and therefore you need to have some understanding in phylogenetics.