“Ah well! I am their leader, I really ought to follow them!” - Alexandre Auguste Ledru-Rollin

The basic answer is that the person is a person who likes working with people and that can provide leadership for the team(s). Characteristics include:

• Intimately familiar with Agile. If you are implementing Scrum, this means you are intimately familiar with Scrum. Kanban, Kanban. And so on. Since the Scrum Master owns the process, they must be intimately familiar with how Scrum works and be able to guide a team to find solutions to help them succeed. This takes experience. Most Scrum Masters come fresh out of Certified Scrum Masters training and jump into the role. The challenge here is that they often do not have an experienced Scrum Master available to mentor and coach them. This means a Scrum Master must be intensely curious.
• Intensely curious: Most Scrum Masters do not start out being a great Scrum Master. In order to provide value to the Team (beyond just administrative) great Scrum Master are always learning about Agile, leadership, people, organizational dynamics, facilitation techniques and so on. The learning is not something you go on a course for 3 years to get. Most of the time Scrum Masters notice an issue, then go out and see what others have learned in a similar situation, so that when it comes up in a team discussion, the learning can be presented as “one possible approach”. This requires awareness of what is going on, but also the drive to learn more.
• Responsible. In most organizations, when someone is given responsibility they are concurrently given the authority necessary for success. Scrum Masters are in a different situation. While a Scrum Master does not assume responsibility for the success of the project—that remains with the team —a Scrum Master does assume responsibility for the team’s adoption of Scrum and practice of it. A Scrum Master takes on this responsibility without assuming any of the power that might be useful in achieving in it.
• Excellent facilitator. Scrum Masters organize and facilitate several of the Scrum meetings. This requires organizational and excellent facilitation skills. Again, these skills will sometimes take awhile to learn.
• Collaborative. A good Scrum Master will work to ensure a collaborative culture exists within the team. The Scrum Master needs to make sure team members feel able to raise issues for open discussion and that they feel supported in doing so. The Scrum Master should help create a collaborative atmosphere for the team through his / her words and actions. However, beyond modeling a collaborative attitude, a good Scrum Master will establish collaboration as the team norm and will call out inappropriate behavior (if not already done by other team members).
• Influential. To be successful a Scrum Master will need to influence others both on the team and outside it. Initially, team members may need to be influenced to give Scrum a fair trial or to behave more collaboratively; later a Scrum Master may influence a team to try new technical practices such as Test-Driven Development or pair programming. A Scrum Master should know how to exert influence without resorting to a command-and-control “because I say so” style. Most Scrum Masters will also be called upon to influence those outside the team. A traditional team may need to be convinced to provide a partial implementation to the Scrum team, a QA director may need to be influenced to dedicate full-time testers to the project, or a vice president may need to be convinced to try Scrum at all. While all Scrum Masters should know how to use their personal influence, the ideal Scrum Master will come with a degree of corporate political skill. Corporate politics is often used pejoratively; however, a Scrum Master who knows how decisions are made in the organization, who makes them, which coalitions exist, and so on can be an asset to a team.
• Highly available to the team. If the Scrum Master has a full time role elsewhere in the company and just runs in for the daily scrum meeting, they are not going to be very helpful to the team. They need to manage impediments and be available as needed to the team to work through any roadblocks. The Scrum Master should always know the status of the team and how things are going.
• Know when to stop talking. This is a tough one. Often times, skills that make a person a good Scrum Master are the same skills that make it hard for them to keep their mouth shut. Scrum Masters need to avoid telling team members what tasks to do in the daily scrum or start questioning the team’s estimates during estimation sessions, for example. As a Scrum Master, you need to know when to keep your mouth shut and when to step in. You are not a traditional project manager and you should not operate in a command and control style. You are a “servant leader.”
• All about the Team. As a Scrum Master, you are always focused on the team, not on yourself. A Scrum Master is humble. You need to thrive in watching the team succeed and do whatever it takes to help them do so. If you need a lot of personal recognition and praise, then a Scrum Master may not be the best job for you. One way to think about this is to look at the way sports coaching works: if the Team wins, it’s because of the Team, if it loses it is because of the coach. A great Scrum Master, like all coaches understand that it is all about success of the Team.
• Sensitive “BS” detector. A Scrum Master does not have to be expert in the areas the team is working on but should know enough to understand when someone is not being 100% honest. A lot of the time the Scrum Master does this by asking questions (“Could you explain that a little more?” or “Could you draw a picture of this to help me understand the issue?”)
• Wants, and understands, the role: Many organizations assign the role of Scrum Master. Many Scrum Masters become the role without really understanding the role. The reality is that to be a change agent in the organization, to work the difficult inter-personal issues is difficult. And if you are being “forced” to do the role, you will not be difficult. So the Scrum Master should understand the role and more importantly wants the role.
• A “growth mindset”: As opposed to a “fixed mindset”. A growth mindset believes that capabilities can be developed, the love challenges, and that treat failures as learning opportunities and can apply this viewpoint to both themselves and their teams.
• Challenges the Team: As a result of the growth mindset, the Scrum Master creates an environment where Teams are challenged to solve problems that they may not have thought they could do previously.

Notes:

• A lot of this text was developed by reading other web sites but it was done a long time ago and so I have lost the references. If you can provide information on sources I would be happy to attribute.
• Scrum Master is obviously a role in Scrum. Increasingly I have found that the role of coach, mentor, teacher, etc is something that other practices can benefit from as well, depending on context. This write-up reflects this more general understanding.

• Grown mindset: https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve