If you happen to have a fallout with someone at work, you will soon see how complicated your working life has just become. Unlike a romantic partner, you can not just break up with someone you work with, unless you are already unhappy in a bad job and wish to leave anyway. If you love your job and don't want to go anywhere, having problems with a coworker can make your life miserable. Though it can be hard, you have to think about how to go about repairing relationships at work. They may never be the same, but things can get better.
How hard repairing relationships at work may be depends on what happened. If you took credit for someone's work that you had no part in, someone you work with is going to feel quite angry with you on a professional level. If you went out for drinks with a coworker and saw them do something embarrassing, and then you promptly told everyone about it the next day, the damage is on a more personal level, but it can have a bad effect on your working environment. Both are tricky situations, but there are some things you can do to turn things around and repairing these relationships.
You have to first realize what you did wrong. You can not go about repairing relationships if you do not realize what you did to someone else was hurtful and should not have been done. You can not apologize and make things right if you do not feel that you did anything wrong. Understand, from your coworkers point of view, what you did and how it hurt them. Once you are truly understanding and sorry, repairing relationships is so much more easy. If you are insincere, it will show when you try to make things right.
If you have stolen someone's ideas, or perhaps fumbled on a project and blamed someone else, you have to approaching repairing relationships at work with humility and with professional calmness. The coworkers in question must be willing to hear what you want to say to them. Start with an email, telling them that you want to talk. Go to your boss and tell him or her what you did and how you intend to repair things. Then talk to your coworkers and be as sincere as possible and offer solutions to make things up to them. It won't always work, but you have to try.
On the other hand, if you have made a professional booboo that involves someone's personal life, repairing relationships won't be quite as hard on you professionally, at least not now. However, you never know where someone else may show up in your career in the future. Talk to them about what you did and how bad you feel about it, and then when you hear the gossip circulating, do all you can to nip it in the bud, even if you have to say you made it up or exaggerated. You may look a little silly for a while, but repairing relationships at work that can take a toll on your career is much more important.