Personally speaking, I have still not quite got my act right on how to apologize but I do know what is the easiest way to do it, read this somewhere. The first thing is to set your ego aside, that is the hardest part of the whole exercise. Rest is downhill, piece of cake and gets easier with every next time.
Here are the most precise and concise words to use. You can translate as per your language of speech.
I am sorry. It was my fault. How can I make it right?
Yes that is right. Simple, straight to the point. A usual follow-up would start a conversation which will lead to eventual acceptance of the apology either said literally or known through how the conversation ended.
Learning to apologize is a very important skill as it helps build tolerance in society and make you realize that you are not the only one in this world. Improves your emotional intelligence and helps you grow and improve as a human being.