To some degree, it is unfortunate that those who run projects are referred to as managers because there is so much involved in being a truly excellent project manager.
Almost all project literature will list the following as, at least, the core and most important characteristics of a project manager:
Ability to influence the organization
The traditional view of management and leadership is that management is concerned with efficiently and effectively using a company's resources to accomplish the company's business, while leadership is more concerned with innovation, challenging the status quo, and broadening the company's outlook and capabilities. Managers try to get people to agree about the things that need to be done. As Warren Bennis and Burt Nannus (professors at the University of Southern California) so succinctly describe the difference between managers and leaders; "Managers are people who do things right and leaders are people who do the right thing." 
Project managers have three basic responsibilities in managing a project: to be on or under budget, to be on or ahead of schedule, and to meet the customer's performance criteria. So it makes sense that project managers must have management skills to accomplish the project's goals successfully. But management skills without leadership skills are very likely to lead to poor results or even failure.
An in-depth study identified a number of characteristics important in a leader.  The top four characteristics were: honesty, competence, the ability to look forward, and inspiration.
Communication skills are important for any manager. For the project manager, they are absolutely critical. When one considers the amount of time a project manager spends communicating with his team, the project sponsor, stakeholders, and senior managers throughout the organization, it becomes readily apparent that a poor communicator has almost no chance of completing the project as planned. The project manager requires not only excellent speaking skills, but writing skills as well. This is because a large percentage of communication is in the form of reports. When I think of how important communication skills are in project management, I am often reminded of a classmate who, when he discovered that he was required to take a speech class, exclaimed that he was studying to be an engineer, not a speaker. In today's business world, there are very few professions in which success does not depend on how well the practitioner communicates.
Negotiating skills rank closely with communications skills in importance, and indeed one cannot be a successful negotiator without possessing excellent communication skills.
Project managers are constantly faced with issues relating to scope, cost, and schedule objectives, organizational objectives relative to the project goals, changes to the scope, resource assignment and allocation, and team conflicts.
Many project managers also are involved in contract negotiations. If a project is the result of an external contract, it is not unusual for the project manager to have been a part of the negotiation team. Likewise, the project manager usually is a part of the negotiation team when hiring vendors, or when teaming with other companies to pursue a project.
Problem solving is more than evaluating a problem and determining a solution; it also involves making a decision.
Project problems can be the result of technical incompatibilities or even the lack of a technical capability. They can be interpersonal in nature or they could result from functional managers reassigning one or more of the resources. They also can take the form of external difficulties with environmental or other regulating agencies. Whatever the source, it is the project manager's responsibility to assess the problem and determine the best course of action to resolve it. Finding a solution to a problem, though, is only half the job. A decision about how or even whether to implement the solution must be made.
Decision making usually involves investigating several options and choosing the best solution for the problem and the good of the project.
W.G. Bennis and B. Nannus, Leaders: The Strategies for Taking Charge (New York: Harper & Row, 1985).
James M. Kouzes and Barry Z. Posner, The Leadership Challenge (San Francisco: Jossey-Bass, 1989).