The word "opponent" focuses on someone who has the same aim as you and only one person can achieve the aim and be the winner. Therefore, there is an element of competition and your "opponent" will vie to achieve the aim before you. This word is commonly used in competitions, fights and games.
Example: The dictator made a brutal attempt to silence his political opponents prior to the elections.
Someone can also be an "opponent" of something if he or she disapproves of it and is trying to prevent it from happening.
Example: He has been a life-long opponent of euthanasia.
An "opposition" can comprise of a group of people or a person who does not agree with a current situation and wants to change it OR wants to prevent a situation from changing. Here, the focus is on action of opposing or the state of being opposed.
Example: The plan to build a highway near the town is likely to meet with fierce opposition.
I hope this helps.