The Latin word "ambasciator" is formed from the verb "ambasciare", which means "to send on a mission" or "to dispatch". This ultimately comes from the prefix "amb-", meaning "around" or "on both sides", and the verb "asciare", meaning "to listen to" or "to obey".
So, the word "ambassador" essentially means someone who is "sent around" or "dispatched" to represent a person or group.