what is the role of Project management in an organization?  

In my experience "management" oversees the running of the entire business. "Project management" is responsible for only certain aspects of the business such as building, advertising, employment.

Project management is about overseeing (or managing) all of the resources that are required to bring a project to completion on time and within the constraints of an allocated budget. A project is a "temporary" task, with a specific start and end. There is usually just one Project Manager, who is basically the "go to" person for that specific project. Management, in and of itself, would be an ongoing responsibility, not linked only to a certain project... being responsible and overseeing all, or a certain part, of a business or an organization.

Mangement is the overseeing of the employees for performance. Project Management is a leader that handles the flow of information back to the Management regarding a project or event that they are assigned. Management then would have the position of more power within a company. Hope that this leads you to what you seek to know.

it is based on following 9 knowledge areas recommended by PMI(Project Management Institute), USA.
1- Time management
2- Cost management
3- Scope management
4- Quality management
5- Risk Management
6- Integration management
7- Precure management
8- Human Resource management
9- Communication management

Management is overall taking care of things. I will give you a small example. A housewife manages her whole house. However if she decides to get her house painted, that is a certain project and she becomes the project manager for that :)

In an organization it is the same. For example My immediate boss is the Manager for a specific area of the organization that handles specific type of projects. He watches over everything that happens in his department. His employees performance, the various projects they are working on etc. he gives me a project to work on and I become the project manager for that project. Eg. setting up a school in a certain area is one project and anyone who's taking care of that project is the project manager for it.

I hope I have been able to explain it.