There are a few different meanings of being official. One meaning is when two people are in a relationship and they make it official by declaring themselves boyfriend and girlfriend, or husband and wife. Another meaning is when an organization, company, or government makes something official, like an announcement or a decision. And finally, when someone achieves a goal or gets recognition for something, you can say they are now official.
For example, let's say John and Jane have been dating for awhile and they tell their friends that they are now an official couple. Or, the president of a country makes an announcement that their country is now at war with another country. In this case, being official means that it is public knowledge and has been made legal or binding.