Question

What are stars that pulsate in and out called?

Answer 1

Stars that pulsate - growing brighter and dimmer - are called variable stars.

Explanation:

There are a number of different types of variable stars. Some are variable because they are close binary pairs that eclipse each other, and some actually cycle in size and brightness. Still others, called cataclysmic variables undergo random or periodic violent explosions.
For more on variable stars, there is a good summary here .