Each team has a large and complex management structure, as the running of a Major League ball club (and its minor league teams) is a huge task. The management can generally be divided into "The Front Office" (the "suits") and "The Coaching Staff" (who look after the players day-to-day, and during the game).
The Coaching Staff
The Coaching Staff are the most visible of the team's management, and are the people you see making the decisions during the game, on the field of play.
The "Manager" has overall responsibility for running the team, and generally will be seen sitting on the team bench. It's the manager who decides which players will play each day, when to rest players, when to make a substitution (whether a pitcher, a pinch hitter or a pinch runner), and when to come out and argue with an umpire (fairly common, particularly for some managers). It's the manager you'll see walk to the mound to remove a pitcher.
The manager will be assisted by a number of other coaches. The "First base coach" reminds any runner at first base of his options, and will warn him about pickoff moves and so forth. The "Third base coach" is responsible for telling runners when to go from second to third, and more importantly when to round third and head for home - he's often very visible "waving in" a runner or "putting up a stop sign" to stop a runner.
Teams will usually also have a "bullpen coach" who is responsible for looking after pitchers in the bullpen during the game (making sure they warm up properly), a "hitting coach" (responsible for coaching the players during batting practice (normally carried out before a game), ensuring they don't lose their technique and so forth), a "pitching coach" (who does much the same for the pitchers) and even a "bench coach" (responsible for looking after players on the bench, and usually a sounding board for the manager).
Minor league teams will also have a manager and coaches, though the lower the minor league, the fewer the number of coaches there are likely to be.
The Front Office
Managing a team day-in, day-out, is a complicated enough task for the manager, and a team will generally have a "General Manager" (or GM) who is responsible for looking after the wider player-personnel issues. When the manager needs a new pitcher, it's the GM who actually decides who to send up from the Minor Leagues, or who to trade for. The manager has to worry about the day-to-day management whilst the GM looks at the longer term picture.
To do both jobs is simply impossible, such that no-one even tries. Some managers become GM's, but the key for any successful club is for the two to work in harmony. It's no use the GM providing the manager with players he doesn't want, and no use the manager not having confidence in players he is provided. More often than not, if there's significant disharmony between the GM and the Manager, the Manager will get fired (or both will!).
The General Manager will generally have a number of assistants, liaising with the minor league affiliate clubs, as well as controlling a large network of scouts who watch other teams' minor league players, as well as college and high school prospects for future drafts.