A core is like the engine of a car. It's responsible for the execution of instructions.
The thing is that both Intel and AMD had reached a point where it was not possible to get more performance out of a single core, without putting in considerable effort. Clock speeds beyond 4GHz are not quite efficient in terms of power draw too. As a result, both companies decided to add more cores to improve performance, rather than ramping up clock speeds
However to take advantage of multiple cores in a processor, software has to be multi-threaded and multi-core optimized.