Re: How can the mutation of a single individual give birth to a new species?

There are several points that you raise in your question, so I'll have to give a bit of a roundabout answer.

If an individual has a single random mutation that gives it an advantage, that individual should be able to produce more offspring than other individuals in the population that don't have the mutation. Those offspring will in turn also be able to produce more offspring than most individuals in the population. Eventually, the gene with the mutation should be present in every individual in the population. This gene is called fixed.

If a single species gets enough new mutations accumulating over time, it would be very different from its original form. This is evolution in a single species; what you end up with (often after a million years or more) is different enough from the original form of the species to be recognized as a different species, even though it was a fairly smooth transition.

One species can split into two (this is called speciation) in a few different ways. One way is to have a population of the species split into to separate populations, often by a geographical feature like mountains or a river. The two populations change over time, as in the single species above. They don't change in the exact same way, as mutations are random things, and so the two populations slowly diverge. Eventually, they may diverge enough that even if they do come together later on, the two populations can't mate with each other and are considered to be two different species. The amount of time that this takes depends on the random mutations; a change in mating behaviour can result in a fairly quick speciation.

Species can also split if it becomes specialized on different food sources; the specialists on the two food sources become so specialized that the groups of specialists start to diverge as the populations above do.

These are all methods that don't happen due to a single mutation, but instead because of many mutations accumulating over a period of time. One way that a new species can spring up almost immediately is when species obtain extra sets of chromosomes from their parents. Usually, you get one set from each parent, but sometimes there is an accident and one parent gives two or something goes wrong in development. In animals this is usually lethal, but in plants it often isn't. Since plants can reproduce without sex and can often self-fertilize, a new species can pop up overnight. This isn't actually a genetic mutation, but it is a sudden process(*).

To summarize, speciation can often happen due to an accumulation of genetic change over time or due to drastic changes in chromosome structure in a very short period of time.

(*)Moderator's semantic note: "mutation" is defined simply as change, and can be any structural change to the genome from a single nucleotide in a single gene to a complete rearrangement of chromosomes. Thus, gaining or losing chromosomes is definitely a mutation.