A compact star is a really, really, really, REALLY small form of a star that is found in outer space. They are so small that they can be even smaller than a city, but they are extremely heavy, like a billion elephants packed into one tiny space.
What makes a star a compact star is its density, which means how tightly packed its matter is. In a normal star, the matter is spread out enough that it doesn't experience a huge amount of pressure. But in a compact star, the matter is so tightly and densely packed together that it experiences a tremendous amount of pressure.
These types of stars are so powerful that they can do things like create gravitational waves and pull in things like light, causing a phenomenon called a black hole.