How do we call a "burst of evolution"?
There are indeed periods of fast evolution and period of evolutionary stasis. The term "burst of evolution" is never used in the literature but below are three common terms when discussing these concepts.

First we have to realize that the term 'speed of evolution' does not mean much. We can be talking about change in phenotype through time (See Punctuated Equilibrium and Evolution and Ecology time scales) or speciation rate (See Adaptive Radiation and Mass Extinction). Also, we have to realize that we can talk about such speed of evolution on a given taxon or for the entire life on earth.

Punctuated Equilibrium

We often refer to the fact that there are periods of evolutionary stasis and periods of rapid evolution as punctuated equilibrium 


  Punctuated equilibrium (also called punctuated equilibria) is a theory in evolutionary biology which proposes that once species appear in the fossil record they will become stable, showing little evolutionary change for most of their geological history. This state is called stasis. 


Here is a representation (from the above wikipedia link) of phenotypic divergence following an extremely gradual view of evolution and an extremely "punctuated" view of evolution



Evolution and Ecology time scales

Since the 2000s, there has been increased recognition that evolutionary processes even matter on a ecological/demographics scale. That is ecological/demographics processes and evolutionary dynamics influence each other dynamically and the old view that evolutionary processes are much slower than ecological/demographics processes is sometimes wrong.

See the post History: Do evolutionary and ecological processes occur at the same timescales?

Adaptive Radiation / rapid diversification

The speciation rate also varies through time. There are "burst in speciation rate" if you like the term. A sudden burst in speciation rate is called an adaptive radiation


  In evolutionary biology, adaptive radiation is a process in which organisms diversify rapidly from an ancestral species into a multitude of new forms, particularly when a change in the environment makes new resources available, creates new challenges, or opens new environmental niches. Starting with a recent single ancestor, this process results in the speciation and phenotypic adaptation of an array of species exhibiting different morphological and physiological traits. An example of adaptive radiation would be the avian species of the Hawaiian honeycreepers. Via natural selection, these birds adapted rapidly and converged based on the different environments of the Hawaiian islands


You can find a list of classical examples of adaptive radiation on the above wikipedia link or at the post Is there a taxa that radiates faster?. If you don't want to be too extreme, you can as well just use the expression rapid diversification.

Mass Extinction

There are essentially two differences between an adaptive radiation and a mass extinction. 1) one is about an increase in speciation rate while the other is an increase in extinction rate. One is about a single taxon, the other one concern most of life on earth.

Because, a mass extinction sounds quite apocalyptic and because we seem to be at the start of a new mass extinction, most people are familiar with this term. 


  An extinction event (also known as a mass extinction or biotic crisis) is a widespread and rapid decrease in the biodiversity on Earth. Such an event is identified by a sharp change in the diversity and abundance of multicellular organisms. It occurs when the rate of extinction increases with respect to the rate of speciation. Because most diversity and biomass on Earth is microbial, and thus difficult to measure, recorded extinction events affect the easily observed, biologically complex component of the biosphere rather than the total diversity and abundance of life.


I want to highlight that when thinking of a mass extinction, many people think of a single event clearing up the surface of the earth in a day but the reality is very different. In fact, a mass extinction can well last thousands of years.