A simple definition for Project Management would be Getting Things Done!
If you look up Wikipedia,
It is the practice of initiating, planning, executing, controlling, and closing the work of a team to achieve specific goals and meet specific success criteria at the specified time. The primary challenge of project management is to achieve all of the project goals within the given constraints.This information is usually described in project documentation, created at the beginning of the development process. The primary constraints are scope, time, quality and budget. The secondary—and more ambitious—challenge is to optimize the allocation of necessary inputs and apply them to meet predefined objectives.
If you look up the definition as per PMI,
It is the application of knowledge, skills, tools, and techniques to project activities to meet the project requirements. The processes fall into five groups: Initiating, Planning, Executing, Monitoring and Controlling, Closing
One of the important factors to keep in mind is that Project Management will result in a “final” deliverable within a finite time span. To accomplish this, the Project Manager will need a range of skills that not only requires technical skills but also people management skills.
Project Managers may use a wide range of tools to aid them in accomplishing the task at hand. In the world of Software, some of the popular tools include Atlassian JIRA (and its suite of products), Rally, Microsoft Project, Trello, Basecamp, etc. If your organization doesn’t have the budget to purchase such tools, a simple spreadsheet can suffice to track the project (at least for “simpler” projects). Choice of tools can also vary depending on the development methodology that your organization prefers using, i.e., Waterfall, Agile Scrum, Kanban, etc.
For any project to be successful, Project Management will be aided by other processes that can cover aspects like Change Management, Risk Management, etc. The number of supporting processes will vary depending on the Software methodologies, Processes and Compliance levels that an organization plans to adhere to.
A good starting point would be to look at the article on Software Effort Estimation.