My understanding of evolution is that genetic mutation occurs in individual members of a species, and they become a new species.
Isn't a definition of species a group of genetically similar organisms that are pretty much exclusive in breeding ability?
You are correct in that the origin for any new species lies in mutations which occur in individuals and are then passed down through DNA.
You are misunderstanding that simply because a mutation exists in an individual that they become a new species. Take Lactase Persistence as a widely studied example in humans. The majority of the world is lactose intolerant after they stop drinking their mother's milk, but a single point mutation results in the ability to consume dairy products into adulthood. It does not make people who are lactose tolerant a new species, but it does open up a potential new nutrient source, which benefits the people who have it and thus it was passed down successfully over other variations.
In addition to that example, the definition of "Species" is NOT exactly defined. There are many variations, including (but not limited to):
1) A group of organisms that cannot interbreed in the wild.
2) A group of organisms with significantly different phenotypes (appearances).
3) A group of organisms with sufficient* genetic divergence.
*There is absolutely no scientific consensus on how much genetic divergence is necessary to create a new species. We share >95% of our genetics with Chimpanzees, so it's not much - it could be as little as a few key genes out of the entire genome.
So where you define the boundaries of species is up for grabs to some extent, but the most common is what you delineated - when a member of one group cannot interbreed with another group.
Wouldn't there have to be a male and female with the same genetic divergence at the same time to make a new species? What is the popularly accepted mechanism for the splitting into a new species in organisms that require sexual reproduction?
Nope. Assuming everything necessary for the gametes to produce fertile offspring is present (aka - the chromosomes aren't significantly altered and the gametes are compatible) then a new species emerges through successive generations.
One mother or father gives their mutation to their children, which pass it on to their children, which pass it on to their children, etc. If the mutation is significantly advantageous it will allow the children with the mutation to better survive in the environment, and over time it could become the dominant phenotype that could lead to the creation of a new species as further mutations build in the new group - further differing the generations of the mother/father from the original group the mother/father were born into. Good examples are the HAR - Human Accelerated Regions which are sets of genes that are pretty much the same throughout all other vertebrates, but startlingly different in humans. The HAR did not create humans, but it's thought the mutations contributed greatly to our intelligence - which allowed other mutations to be taken advantage of (like the humble opposable thumb) as our lifestyle changed due to the increased intelligence.
Not all mutations need to be advantageous, though. There's lots of fuzzy areas. If a mutation isn't beneficial, but also isn't that detrimental, it can also stick around in a population. These mutations might not lead to the creation of new species, but can certainly contribute to genetic divergence.