Your coach is an important person responsible for training and inspiring your players.
Skill: A skilled coach knows the best way to train players in order to make them better. The more talented your coach is, the more effective training will be. The maximum a coach can be is excellent. All new teams get a passable coach to start with.
Leadership: Apart from running training sessions, your coach is also an inspirational source for your players. In general, your team spirit will be better if your coach is a good leader.
Tactics: Your coach can be offensive, defensive, or neutral, which (only) affects your team's performance during the game. An offensive coach improves your attack at some expense to your defense and a defensive coach improves your defense at some expense to your attack.
A defensive coach improves defense slightly more than an offensive coach improves attack, to the same expense. A neutral coach is neither offensive nor defensive, which means he doesn't help or hinder any sector.
After your coach has been in charge for one season, his leadership skill will start to slowly deteriorate. Once your coach reaches disastrous leadership, the deterioration starts affecting his trainer skill instead.
If you are not satisfied with your coach, you can either recruit one externally or make one of your own players coach. You can do this from "Training", under the “My Club" menu.
Remember, all new trainers of a certain skill level are equal. If you hire a coach with passable trainer skill for example, he always becomes "average" passable (an excellent coach, however, is only 1/2 step greater than a solid coach).
When you hire a new coach your old coach will remain in the team as a player, but he can never become a trainer again and you can not sell him either. If you don't want to keep him, you will have to fire him.
Recruit an external coach
When you recruit a new coach you first decide what style of play you want your coach to implement, and then you decide how good you want him to be. Each combination of leadership and skill has a fixed price - the better the trainer the higher the price.
Make a player coach
When you turn a player into coach he will keep his current leadership. His experience determines how skilled he can become, and also how much his coaching course to reach a certain skill level costs. So, if you have a very experienced player with high leadership, this can be a real bargain. The player must, however, have been in your squad for at least a season (16 weeks) before you can make him a coach.