Gradualism and punctuated equilibrium are two ways in which the evolution of a species can occur. A species can evolve by only one of these, or by both. Scientists think that species with a shorter evolution evolved mostly by punctuated equilibrium, and those with a longer evolution evolved mostly by gradualism.
Gradualism is selection and variation that happens more gradually. Over a short period of time it is hard to notice. Small variations that fit an organism slightly better to its environment are selected for: a few more individuals with more of the helpful trait survive, and a few more with less of the helpful trait die. Very gradually, over a long time, the population changes. Change is slow, constant, and consistent.
In punctuated equilibrium, change comes in spurts. There is a period of very little change, and then one or a few huge changes occur, often through mutations in the genes of a few individuals. Mutations are random changes in the DNA that are not inherited from the previous generation, but are passed on to generations that follow. Though mutations are often harmful, the mutations that result in punctuated equilibrium are very helpful to the individuals in their environments. Because these mutations are so different and so helpful to the survival of those that have them, the proportion of individuals in the population who have the mutation/trait and those who don't changes a lot over a very short period of time. The species changes very rapidly over a few generations, then settles down again to a period of little change.
This explanation talks about punctuated equilibrium as the result of one or a few mutations that cause large change. However, punctuated equilibrium is any sudden, rapid change in a species and can also be the result of other causes, such as huge and sudden changes in the environment that result in more rapid changes in the organisms through harsher selection.
How did the tiger get its stripes: gradualism or punctuated equilibrium? We don't know whether the tiger got its stripes through gradualism or punctuated equilibrium, but in order to explain both concepts, here is how it could have happened through each. Let's assume that stripes are helpful because they help the tiger to camouflage, blend in with the tall grasses where it lives, so that it can sneak up on its prey (what it eats) and not be noticed.
Gradualism: A long time ago, there were a lot of tiger-like animals, but without stripes. Most of them were unmarked, but a few had light markings and color variation in their fur. These few blended in with the tall grasses a little bit better, so they were generally able to catch more food, and fewer of the marked than unmarked ones died of hunger, so more of them were able to reproduce. In the next generation, more animals were marked than in the previous generation. Of those that were marked, some had more, some less, and some the same amount of marks than in the previous generation. Also, the marks were more, less, or the same amount clearly defined. Again, the ones with marks did better than the ones without, and the ones with more, clearly defined marks did better than the ones with fewer or fainter marks. Very gradually, over many, many generations, stripes over the tigers' whole bodies formed and appeared in the whole population, because the tigers that survived in each generation were those whose marks were most clear and contrasted most with the rest of the fur, and those that covered the most area on the bodies of the tigers.
Punctuated equilibrium: A long time ago, there were a lot of tiger-like animals, but without stripes. One time, a mutation occurred in a few of the animals, causing a huge change: they were born with stripes! This was so helpful to survival that out of the whole population, none or almost none of those with stripes died of hunger. They lived to reproduce, and their striped offspring also did very well. Over only a few generations, the whole population was born striped.
A combination: Here is one idea of how tigers could have gotten their stripes by both gradualism and punctuated equilibrium: A mutation had a huge affect, causing distinct, stripe-like markings. These were then gradually "polished up" into stripes.
The idea of punctuated equilibrium originated long after the idea of gradualism. Darwin saw evolution as being "steady, slow, and continuous". Later, scientists were studying fossils and they found that some species have their evolution almost "mapped out" in fossils. For others they found a few, very different species along the evolutionary course, but very few or no fossils of "in between" organisms. Also, when dating the fossils, scientists saw that in some species change was very slow, but in others, it must have occurred rapidly to be able to produce such change over such a short amount of time. The scientists reasoned that there had to be another way that evolution could have happened that was quicker and had fewer intermediate species, so the idea of punctuated equilibrium was formed.