If you have offended a person or expressed harsh words about anything, you should definitely apologize. They need to be heard by the unjustly offended person, and it will become much easier for you when you say these simple words.
Put yourself in the shoes of the person you hurt by word or deed. Think not about what you would like to say or explain, but about what this person is waiting for. For example, if someone accidentally pushed you, you don't need to know why it happened, it's more important for you to hear that the person is sorry. Of course, if the circumstances are more complicated than an ordinary confrontation on the street, you need to tell what moved you, but you can do it after, when you have received a principled agreement to forgive you.
Choose a cute postcard with the words "Sorry" or "I am sorry", write warm words in it. This option is especially suitable if a quarrel occurred between girlfriends or teenagers. If you can't find words, just draw a heart on the card and sign. Once the card is received, your chances of engaging in constructive dialogue are increased.
Try not to use high-flown words, if you are addressing a loved one, you will hear much faster "I'm sorry", "Forgive me, please." But in business relations, when it is necessary to clarify an unpleasant moment, you can choose "We are sorry", this is especially appropriate in the correspondence between organizations.
Be sincere. If you do not consider yourself to be guilty, and ask for forgiveness only because it is so customary, your words may seem fake to the offended person, and this will only aggravate the situation.
Try not to touch the person until you can tell from their reaction that they are ready to forgive you. This can be perceived as a desire to impose peace or penetration into personal space.
Remember that some people really like to be offended and believe that they have been treated unfairly. Try to express your apology as honestly as possible and leave the person for a while. The fact that you asked for forgiveness already softens your guilt in many ways. The person will cool down a little, think and then take the first step.
Even if you received forgiveness, try to analyze what exactly led to the conflict situation and not repeat these mistakes in the future.