From the time you were small you were likely assessed on how you relate to others around you. Once you enter the working world, "plays well with others" graduates to "works well with others" and is a critical indicator of your future success. In everything you do you will need to depend on or work with another individual, so this skill is really rather non-negotiable, even though some will need to work a little harder at acquiring it. Whether you are the boss at the top of the ladder or the grunt at the bottom, your willingness to be part of something bigger than yourself will measure your true success.
A Link in the Chain
The minute you are hired into a company you become part of a team. Even if your job is highly specialized and you do most of your work alone, your duties are a link in a much bigger chain. "Working well with others" means you give equal respect to those around you and their equally important position in the chain. Learn how what you do helps others, and don't be afraid to ask what you can do to improve to make things more efficient and streamlined. This means learning other tasks so you may help as needed, rather than pass it off as "not your job."
No "I" in Team
For those projects or positions where you work closely with a larger group of people, the definition of "working well with others" will describe your willingness to be accommodating and share equal responsibility for a shared task. Be as willing to take responsibility for things that go wrong as praise for things that go right, and never try to hog the glory. Respect the contributions of everyone on your team, and welcome collaboration.
The Golden Rule isn't just for Sunday School. You should always show respect and kindness to those around you in business and in life. Never spread rumors, insult others or sabotage your anyone with passive aggressive behavior. Instead aspire to be a person of integrity. Honor your word and fulfill your obligations. This is easier to do when you genuinely like your coworkers and you honestly want them to do well, but extend this kindness and simple human courtesy to everyone. This creates a positive work environment which encourages others to grow, which benefits everyone.
An old saying suggests to be a true leader you must first know how to follow. "Working well with others" applies to everyone, even the boss. Entrepreneur lists having a strong team as one of the 10 characteristics of a strong leader. Realistically, you will not be able to do everything as your business grows or as you climb the corporate ladder. You are only as strong as the team around you, and they deserve your respect. If you treat anyone under you with disrespect, they may not feel as invested in your success, which only hurts you in the end. Encourage them, ask for their input and be willing to roll up your shirt sleeves and jump in if you are needed.