The pitching designations represent how you, the manager, would like that pitcher to be used, which may or may not matchup to how the pitcher was used in real life that particular season. Here are the general guidelines:
Rest: Only use this pitcher as a last resort
Mopup: This pitcher should only be used in blowouts (either winning or losing). Blowout is determined by the deficit (or lead) and the inning, e.g. an 8 run deficit in the 2nd inning would not considered to be a blowout, but the same deficit in the 8th inning would be.
Long Relief: This pitcher would be called on in a situation in which the starter is pulled early in the game (6th inning or earlier). A long may also be used later in the game depending on the situation and who is still available in the bullpen.
Specialist : These are pitchers who are almost solely used just to get the opposing team's most feared hitter out late in a close game. For example, a left-handed specialist will enter the game to face the opponent's best left-handed hitter (and vice versa).
Setup: This pitcher will be primarily used in the later innings, 7th on depending on the situation.
Closer: This pitcher will be primarily used in the 8th and later innings, depending on the situation and your settings.
The A/B designation is an additional control for setting pitcher usage. An A designation will almost always be used before a B designation of the same role, e.g. if you have one pitcher set to Setup A and another set to Setup B, the Setup A would receive priority in a setup situation. If both pitchers are set to A, then the SimManager will decide who to use based on the situation (opposing righties/lefties, pitcher fatigue, etc.).
*** Please note that if you don't designate a pitcher for a specific role, you may be sending the wrong (or right) message to the SimManager. For example, if you put all of your bullpen pitchers on rest, that would be an indication to the SimManager to leave the starter in as long as possible, regardless of his call bullpen and pinch-hitting settings. Another example might be setting all of your pitchers to rest except for a closer. This would indicate to the SimManager to leave the starter in until a save situation arises. If the starter is getting shelled in the 5th inning, the SimManager would leave him out there rather than use a closer in the 5th inning. It's recommended to have at least 1 pitcher set for each role. Not having a pitcher available at a specific role may have undesired results.