CNN.com recommends starting by owning up to your actions immediately. Without doing so, you won't be able to ask for forgiveness from the individual. When talking through the issue, make sure to directly address what the problem was - this will let the person know that you understand why he or she is offended.
Once you acknowledge your wrong-doing, you can explain that you're sorry. It's important to keep in mind that the person might not immediately accept your apology - be patient as he or she allows your apology to soak in.
If your conflict is causing issues in the workplace, consider speaking to your psychic advisor for advice. A phone psychic reading can help you see the situation from your colleague's shoes. Knowing the individual's feelings on the matter can give you more insight into how to prevent these problems from occurring in the future.