"On the cob" describes corn still on its central woody core, with the kernels intact rather than removed - as opposed to canned or frozen kernels that have been cut off. The cob itself is the tough central spike the kernels grow around; it's inedible on its own, though it's often left in place for cooking and removed before eating.
Corn on the cob is typically boiled, grilled, or roasted after the husk (the green outer leaves) has been removed, then eaten by hand, kernel-side first, or scraped from the cob afterward. The dried cob, once the kernels have been removed, is also useful in the kitchen: it can be simmered in stocks and soups to add corn flavor, or used to smoke meats.
In Mexican cooking, grilled or boiled corn on the cob known as elote is a popular street food, seasoned with lime, chili powder, cotija cheese, and mayonnaise; whole cobs are also a traditional ingredient in pozole. In the American South, boiled or grilled ears are sometimes called "roasting ears." Beyond the kitchen, dried cobs are used as animal bedding, a mushroom-growing substrate, and even fuel.