When taking an Oath you raise your right hand and hold it up until you're done taking the oath which usually takes less than a minute.
While holding your right hand up you also usually put your left hand on a book such as a law book or bible and if it's in court you may not have to swear an oath on a bible or book.
But if you get sworn in as a law enforcement officer or President, Mayor etc then you may need to have one hand on a book and the right hand in the air.
When you take that oath you must really mean it as well because if not and you do something against that oath that you took then you could get into trouble and sometimes serious enough trouble to go to prison or jail.