You can assign your staff to a particular role at your club. If you think your staff is good enough to handle multiple roles, you can also change his role at any time you wish.
Coaches help your players with their training. There are multiple training programs available for your players (discussed in Player Training section), and the best coach available at your club for the given training program will handle that program.
Scouts are used to understand a player better, both your own players and players owned by other clubs. Players have many hidden attributes and without proper scouting you cannot judge a player well. A good scout will give you accurate scouting reports about a player's hidden attributes and potential, so you can decide how to utilize your player better or even make transfer decisions.
Physios take care of the health and fitness of your players. Good physios help your players to regain their fitness faster after matches or even recover from injuries faster.