Your coach is an important person responsible for training and inspiring your players.
Skill: A skilled coach knows the right methods to train players in order to make them better. The more talented your coach is, the more effective training will be. A coach can never be better than excellent. All new teams get a passable coach to start with.
Leadership: Apart from planning training activities, your coach is also an inspirational source for your players. Your team spirit will be better in general 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 the expense of your defence and a defensive coach improves your defence at the expense of your attack.
A defensive coach improves defence slightly more than an offensive improves attack, to the same expense. A neutral coach is neither offensive nor defensive, which means he doesn't aid any team part or incur any penalties.
After your coach has been in your club for one season, his leadership skill will start to slowly deteriorate. Once your coach reaches disastrous leadership, the deterioration starts hitting his trainer skill instead.
If you are not satisfied with your coach you can either recruit an new external one or make one of your players coach. You change coach from the training caption, under "your club" menu.
Remember, all new trainers of a certain skill level are equal. If you purchase a coach with passable trainer skill for example, he always becomes "average" passable (an excellent coach is however 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 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 tactical type of coach you want, and then you decide how good you want him to be. Each combination of leadership and skill has a fixed price, the better trainer the higher 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 the 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 a season (16 weeks) before you can make him a coach.