Punctuated evolution refers to a short (in geological time) period in which a burst of evolution occurred with either a species rapidly changing into a different species or a bunch of species evolving from one species. It is stated that this goes against the theory of gradualism proposed by Darwin.
To my mind, this is hardly surprising, and Punctuated evolution is happening all the time. The real surprise, the thing we have to explain, is why species stay the same for long periods.
The reason is that punctuated evolution is overlayed/restricted by natural selection and natural selection is much more involved in keeping species the same rather than creating new species. Ever since Darwin we have been focused on the role of natural selection to create new species. Examples of punctuated evolution are all around us.
Tale the Wolf/dog situation. If we were looking back on the fossil record from half a million years in the future, of the most recent 30,000 years, when man has apparently being associated with the wolf/dog, we would say that the wolf had experienced a period of punctuated evolution resulting in a vast number of varieties of dogs. 30,000 years is just the blink of an eye in evolutionary terms. We would also be aware that, if any of these varieties had become geographically isolated (as may well be the case if we destroy our present technological society and have to revert to walking to get anywhere) then by simple genetic drift and the accumulation of different genes, genetic cross overs and flipping of sections of genes, these varieties would no longer be able to produce viable offspring (in other words would have become separate species). Looking at the fossilized skeletons of these varieties, a future paleontologist could be excused for positing that the skeletons of a Chiwawa and a Great Dane were from different species
What this indicates is that the wolf species (and virtually all species) are throwing out sports (mutations) at a rapid rate*. So, now we have the question, why hasn't the Wolf in nature radiated into a whole bunch of different forms. This is where natural selection comes in. Once a species is pretty well adapted to a reasonably unchanging environment, natural selection, will work against new 'sports' that are for the most part more poorly adapted to the environment than the dominant form.
*And the rate of the production of sports is far larger than even the variety of dogs suggests. Add to the apparent number of sports that man has selected for to produce dogs with characteristics different to the wolf, to the number rejected and it is amazing that Natural Selection has kept the Wolf pretty well the same over this period (and before) .
The fealty of the wolf to its body plan is less difficult to grasp when we realize that a female alpha wolf in nature can typically produce 5 cubs each year and typically she can live for 7 years. That is 35 cubs produced during her lifetime. But,we are not over-run with wolves. Once population equilibrium in a new area has been achieved, a pair of wolves will leave, on average 2 offspring. Thirty three have been culled by nature*. Natural selection is a bitch. Random accidents will kill some but basically, only the wolfiest cubs will survive to reproduce. And almost every mutation to an already highly successful wolf will put that wolf at a disadvantage.
*Sorry, I have played fast and loose with Wolf biology. In actual fact, in a pack, only the partner of the Alpha Male will produce cubs. Let's say that an average wolf pack consists of 10 individuals. Then you could say that 10 wolves produce 35 cubs over 7 years. 10 survive to replace the pack and 25 have been culled by nature. That is still a harsh level of selection against sports surviving plus the fact that the other wolves are likely to kill a less wolfish wolf. Note that wolves are able to have pups at 10 months of age although this is unusual. But this adds to the potential number of wolves produced by the original pack of wolves and on average, 10 wolves only leave 10 wolves to carry on the species.
Lets look at the somewhat trite example of the color pattern of the common pigeon. The natural color pattern is that dun grey banded pattern. Suppose a white pigeon is produced. As soon as it can fly it is the target of every peregrine falcon in the area. It is far easier to focus on an animal of a different color in a flock. Same for a stripe-less zebra in a herd of stripy zebras.
You might ask why in the park when we feed the pigeons do we see all sorts of other colors. This is because man has bred pigeons for thousands of years both for their meat (Squab) and for their homing characteristics. And we love to save sports and breed from them. When you have a flock of pigeons, you habituate them to a feeding time and then, after a while, release them to fly just before the feeding time. These birds of different color patterns "pollute" the wild birds.
Suppose we went to an island far from the mainland and bred white pigeons. Once we had a large flock established, we brought in Peregrines. I bet they would cull any non-white birds that appeared.
So when will punctuated evolution occur. There are a number of instances but in essence it is when the environment changes.
This could be when a species finds a new geographical area where its kind is not extant. There is lots of room, for a while, for sports to get away from the flock/herd
Or it could be when a natural disaster has altered the environment by physically altering the landscape or by eliminating many existing species.
A key thing is that a disaster has not altered the environment so much that it is outside of the ability of the species in question to survive. So where do the variations come from that allow a species to branch out and survive.
Of course, there are new mutations but will one come along in time to save the species. Undoubtedly this is possible and will have happened in the few billion years that organisms have existed on earth. However there is another source.
Any population will have been accumulating mutations over its existence. They may have altered some members of the population within the envelope of survivability or they may be recessive genes. That is to say, ones that are not expressed phenotypically (visibly) unless two of them come together in the same individual. Very successful species have lots of variability in their genes (look at humans), other species are very specific in the conditions they need (Panda and Koalas for instance). Both need very specific foods, Humans can get along with almost anything organic as nutrient.
The Galapogos Finches are an interesting example of very fast evolution from existing genes within their genome. As we shift from El Nino to La Nina and back, some of the islands produce a lot of large hard seeds and then a lot of small easily cracked seeds and back again. The beaks of one of the species of finches follows these trends with different beakes. All this really means is that both variations of beakes are coded into the genome of these finches and year by year, the ones that have the greatest number of young with the suitable beakes survive in much larger numbers.
Sparrows give us an interesting example. A successful pair of sparrows can produce 5 young, five times per year and live for over 5 years. Yet we are not overrun by sparrows. Their population remains about the same from year to year. Just think what the attrition rate from natural selection is on sparrows. Successful sparrows have to be top notch in all sparrowy characteristics to leave offspring.