Is human race an example of variation? Or is different kinds of dogs an example of variation?
There is only one human race,
Homo sapiens, but it shows many thousands of individual differences: those are variations.
You may be confusing "variation" with "variety". Same root, but slightly different meaning.
When Darwin spoke of variations, he was referring to those differences which distinguish one individual from another. These are the kinds of differences where mutations have a role.
But "variety" or "sub-species" refers to a group, not to individuals. These are groups which are identified by common
similarities not differences. Within these groups, you still get individual differences or variations.
To summarize: variation refers to differences in a character trait such as colour of hair (black, brown, blond, red). Variety refers to a group in which all members share certain similarities. So if I say Greyhound or Chihuahua, you have in each case a very definite image of one variety of dog, and you know that all dogs of that type share many features in common. Yet if you were to examine many Greyhounds or many Chihuahuas you would still find variation within each variety, as you still get many individual differences.
Or do we simply "discover" variations?
We observe variations. Some are easy for anyone to see: hair colour, eye colour, shape of nose, whether or not a person has Asian eyes. Others call for tests to be made--like blood samples to determine blood type. Others can be seen with careful study with a microscope because the differences are small and subtle (how many hairs on a fly's leg?)
What we may then discover are which genes are responsible for the characteristic and how differences in the genes account for differences in the characteristics. You are probably familiar with the term "allele". An allele is any one of the various versions of a gene. The individual differences we see in any population are due to them having different alleles for certain characteristics. So, in the pea plants he studied Mendel found that there were two alleles for seed colour. One version (allele) of the gene for seed colour produced yellow seed. A different version produced green seed.
How do mutations fit into this? Well, why do genes show differences? Why do we have many different alleles of the same gene? Mutations are the answer. If you think of a gene as being a sort of recipe for making a protein, then if you get a change in the DNA sequence of a gene, you make a change in the protein, and that means it may act a little differently and you get a different result--like a new seed colour or a new eye colour, or a differently-shaped nose, or whatever.
So, to sum up: we see variations (individual differences) because genes vary. And genes vary because their DNA sequences are changed i.e. they mutate. And this is not just theoretical. There have been many, many experiments showing both of these statements are true. Genes mutate, and changed genes produce variations that distinguish one indivdual from another.
But what is important at this stage is to note that we are talking individual differences, not group differences. Getting to group differences is another chapter.
Have we really observed that mutation leads to stable variations?
I think what you really mean here is "variety" (See above). This is part of the next chapter. How do individual differences contribute to forming group differences. There are a number of new factors we have to bring in here. Mutations alone do not suffice to explain this. Mutations account for individual differences, but for a stable variety to form, you also have to develop a set of similarities.
I remember the experiment on E. Coli a while ago and it seems suggested that after the mutation of thousands of generation, a variation of E. Coli was finally observed in lab. Is that a good example? E. Coli is not even an animal.
E. coli is a species of bacterium. Even before the experiment began, there were many varieties of E. coli. Here is a sampling of them:
http://www.ncbi.nlm.nih.gov/Taxonomy/Browser/wwwtax.cgi
That experiment was not notable for turning up a new variety of E.coli---since there were already plenty of those. In fact, they began with 12 different varieties and had been documenting various changes that had occurred over 30,000 generations. What made this particular change significant was an unusual innovation shown in this particular variety, and the evidence that it was due to a building up of later mutations on top of earlier ones.
http://myxo.css.msu.edu/lenski/pdf/2008, PNAS, Blount et al.pdf
This is certainly a good example of how mutations change character traits.