Well, for each one..
Player:
Go to a player's information screen, and go to personal, and you'll see reputation.. The best players in the world will have an international reputation and be known throughout the world, while players in low leagues will have a local reputation and be rather obscure. As a player improves or plays for bigger clubs, their reputation will increase.
Club Reputation:
Same thing. Go to the club information screen and look at the reputation. The better your club becomes, the higher their reputation goes. It's also affected a lot by which country you play in. Obviously, a good team in Portugal is going to be a little more known than a good team in Ukraine.
Competition reputation:
Like Jim said, this will usually stay the same. A good example is the FA Cup / League Cup. The FA Cup is a very storied competition, and most clubs would obviously prefer to win the FA Cup over the League Cup. Same with Champ's League and EURO Cup. The Champion's League has more of a reputation for excellence than the Euro Cup, so if you win the Champ's League, your board would be happier than if you won the EURO Cup because Champ's League is more reputable.
National reputation:
A nation like England is going to have a good reputation because the national team is world class (wait a minute..) and the domestic leagues are of good quality. This can change, because a nation like Canada or so could go from a lowly footballing nation to winning the World Cup, thus their reputation would increase.
To view a reputation change, you could use a FM Genie type program to see the actual reputation number in the database as it changes, or just keep track of what the information screens say (ie a club's reputation changing from local to regional)
I'm not entirely sure if I answered your question, and if I didn't just let me know and I'll be glad to help.
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