Very good. Let's start some learning (at least on my part). Be mercy on me as I am pretty slow on biology. If you used too many jargons, I may quit read it.
According to the idea of PE, speciation could take place pretty fast ( <10E6 yrs). And for some strange reasons, at a particular time in the earth's history, MOST species just started to evolve like crazy for a short period of time (10E6 yrs) and then quit the change for a longer period of time (10E8 yrs). Is this concept right?
If so, what caused this burst of evolutional change? Yes, there are various reasons, such as isolation, environment, etc. etc. But these should all be consequences of a more fundamental change. What is that?
Depends if you are actually eager to learn... I'll give this one go...
According to the idea of PE, speciation could take place pretty fast
Yes. The major legacy of PE is that we understand evolution doesn't happen at a uniform rate (some of the more extreme claims of PE have been roundly refuted by evidence).
In the 35 years since PE was put forward, this has been shown in many contexts as well as the fossil record (which is what SJG+Eldridge were originally drawing from).
quit the change for a longer period of time (10E8 yrs). Is this concept right?
No, not really. There would still be observable genetic change (particularly in non-functional parts of the genome*).
The question is a bit backwards.
Evolution isn't a process independent of the things it is acting on. Darwin's key insight is that it happens in response to competition for a biological niche (where niches are in turn functions of the other species in the environment - hence co-evolution).
Over large periods of time the environment can remain relatively stable, so a balance arises between the species forming the ecosystem in that environment. (In game theory this would be called a Nash Equillibrium: the math of coevolution is very similar to the math of game theory).
We know that it is _very_ difficult for evolution to improve on species that are highly adapted to their niche. Evolution is quite a fast process when a species has a clear optimisation path, however, particularly with very large populations (and evolution predicts that populations should normally exist at their Malthusian limits). But it only goes as far as to find a local maximum on the fitness landscape: in systems theory terms it is a hillclimbing algorithm. In some cases species can enter direct 1 on 1 arms races, but even this can only go so far: eventually the resources expended on the arms race will have to balance the ability of individuals in a species to function as organisms.
So this equillibrium holds for a while (this is the equillibrium in PE, it isn't the same as stasis).
Eventually something changes. Either the environment changes, or one species does find a new adaptation that improves its chances. Often environmental changes are gradual, and it can be a long time before the ground rules are sufficiently different for a new set of adaptations to be favored.
What happens then is that there is a domino effect: old niches disappear and new niches form, and species flood into those niches. The fitness landscape deforms, and the hillclimbers of evolution rush to the new peaks. These cascades of change provide the context for increased likelihood of speciation events. But eventually equillibrium returns.
Now, the interesting thing about this is that I've said all this in narrative terms (pitching it roughly at a good high-school student level). But modern evolutionary theory, particularly genetic evolution, isn't narrative. The substance of the narrative I've tried to express is entirely mathematical, and quantitative. We can build models, run the math and work out exactly what we would expect to see.
And, if you haven't guessed already, co-evolutionary predictions are born out by quantitative measures of genetic evolution.
There are other features of evolution that also add to the jerkiness, but that don't depend on co-evolution. But they are less significant for the overall dynamiucs of evolution: On their own they don't account for things like mass correlations of speciation events in the immediate aftermath of a new adaptation or major environmental change, for example. My specialism is a case in point here: there are features of the genome that speed up evolutionary change when it is needed, but allow stability when it is not. They don't explain the PE itself, but they do show that the genome is adapted to its rigours (this is known as evolvability in the literature).
When Punk Eek was first proposed the prevailing (and to be fair tacit) thought in evolution was that the rate of species change was search-limited: in other words species changed as fast as they could find new improvements. PE raised questions, that received research and answers, and we understand now that the rate of change (at least phenotypically) is niche-limited.
Much of the hyperbole around PE has arisen because people think that SJG was challenging the foundation of evolutionary theory (he would have been horrified [edit: thinking again, more likely highly amused] by the idea). In fact he was pointing out a pattern in the evidence that (while completely fitting into the theory) wasn't directly predicted by the theory. Notice the direction of the implication there: you can't say that because it didn't predict the pattern that it predicted a different pattern - the theory was just not detailed enough to have an opinion either way.
Please don't assume that I've told you everything about this, there is much, much more to know before you could hope to make sensible predictions based on this theory, or even evaluate evidence in its light. I studied this stuff full-time for 4 years before I got to grips with it properly, so I can tell you for certain that 500 words on a forum is barely scratching the surface.
---
* On a slight aside intronic drift is a feature that almost seems designed [

] to allow evolution to confirm its predictions.
A large proportion of your DNA codes for nothing. And even those bits that do code for something tend to be split into separate chunks. When your DNA is transcribed in the nucleus of your cell a complex series of proteins and enzymes graft together bits of genes from various sections of your genetic material. Lots of the DNA never gets turned into anything. It gets ignored or trimmed out.
Because there is no evolutionary pressure [this is a big simplification, but the reasons it is are quite complex] on this DNA (it doesn't affect the creatures ability to survive and reproduce), it accumulates mutations at a reasonably regular rate. We can use it, for example, to find matches between individuals for family and race.
We can compare these bits of intronic code between species. And we find that species that we suspect of having a recent common ancestor have less differences in this code, and those having a more distance ancestor have more differences [actually there are many, many other ways to do this kind of comparison, not just introns, but still - I'm talking about introns here].
Not only that but by calculating the differences we can find the age of divergence (because we know from measurements the rate of mutation) and the intronic dating of those common ancestors corresponds very well with datings from the fossil record.