The Catskill Mountains are located in the southeastern part of the state of New York, in the Hudson Valley region. They are actually physiographically related to the Appalachian Mountains, which extend southward into Virginia and Tennessee, and they are also connected to the Poconos mountains of Pennsylvania. There is a large park in New York called Catskill Park, consisting of some 700,000 acres, and the Catskill Mountains are considered to be the mountains within it or very near its borders. The Catskill Mountains are located so close to New York City—only about a hundred miles away—that they have served as a popular vacation destination spot for city dwellers since the mid-1900s. The highest peak is called Slide Mountain, and it clocks in at an elevation of nearly 4,200 feet. You can find some good maps at the link I've pasted below.

The name "Kaatskill Mountains" is simply another name for the Catskill Mountains, also known as the Catskills. This large mountain range lies in the middle Hudson Valley region of New York State. The mountains take up a fairly large land mass that spans five counties, about forty miles south of Albany, on the west side of the Hudson River, and extend to about a hundred miles above New York City. The town of Sleepy Hollow, where Washington Irving's short story "The Legend of Sleepy Hollow" is set, is in the northernmost part of the mountain range near the ski resort town of Hunter, NY.

This older name is still in use in the region, for example, the Kaatskill Mountain Club is an exclusive winter resort for sports like skiing. 

The area was settled by Dutch explorers, and the name "Catskill" is Dutch and means "Cat's Creek." It may be named for the large predator species of cats that once roamed the region but whose populations have declined in recent years.