A competition is where two or more people try and be the best at something. The 'something' can be literally anything that can be judged on some critera of quality, like writing a story, running a race, baking a cake, or growing vegetables, like in the picture.
There is usually some kind of prize for winning a competition.
A score is a record of the number of points, goals or similar made by one side during a game or competition. What you keep score of depends on the game being played. Scores are usually kept track of on a scoreboard, with a score for each of the teams shown on the board.
A trophy is something you are given when you win something like a competition. In war or when you are hunting, a trophy is something you take to prove your victory. In hunting, a trophy might be the head of an animal you killed, and in war it might be some possessions from the people you conquered.
You win something when you are better or faster at it than everyone else. You win a race when you cross the finish line first. You win an art competition when your picture is better than everyone else's. Your team wins a game like soccer if they score the most goals. And you win an eating competition if you eat more than everyone else.
The opposite of win is lose.