The judge call: who does what, rulings, appeals and what a judge may say
Almost every judge call starts with a player, so begin there. A player is responsible for keeping a clear game state and for calling a judge's attention to any rules or policy violation in their own match (TR 204.7.b.2 and TR 204.7.b.4). They may pause at any point in a match to flag a floor judge (TR 412.1). A spectator is different: they must stay silent, they may alert a judge to a possible problem, but they may not tell the players about a violation or give strategic advice (TR 204.8.c.1, TR 204.8.d.1 and TR 204.8.d.2). At Casual and Competitive OPL (Organized Play Level) a spectator may ask the players to pause while they alert a judge (TR 204.8.c.2), but at Professional OPL a spectator cannot even ask for that (TR 204.8.c.3).
The judge who arrives is a neutral arbiter and enforcer of policy and rules (TR 701.1.b). Floor judges answer questions about rules, card interactions, Watcher wordings and derived game state, and they rule on violations brought to them (TR 204.5.a and TR 204.5.b). The most important habit is when not to act. Judges deal with errors that have occurred and do not stop errors from happening (TR 701.1.c and TR 204.5.c.1), so a judge who sees a player about to move a unit to a battlefield that already holds units from two other players (rule 144.4.a.1) lets it happen and then fixes it. Judges should not intervene unless they believe a violation has occurred, a player asked for assistance, or they observe unsporting conduct (TR 701.1.d), and unsporting conduct is the one case where they may intervene proactively (TR 204.5.c.1). Missed triggers, for example of a conquer effect (rule 383.4.c), are the one error a judge does not step in on, even if they see one (TR 204.5.c). Errors are assumed to be unintentional (TR 701.1.a).
What may a judge say when asked? At high OPL they may not answer a direct question about the result of an action the player has not yet taken. The player has to ask about the rule instead, such as whether Shield adds Might to a unit that attacks (rule 814.1.c), and the judge may ask them to rephrase (TR 204.5.a.1). At low OPL a floor judge may help players play in a technically precise way, and that help should generally not include strategic advice (TR 204.5.e).
Here is a worked example. A player pauses play and calls a judge. In practice the judge asks both players what happened, rules, and fixes the game state; that order is ordinary procedure rather than a rule. If a penalty applies, the judge explains the error, the procedure to fix it and the penalties for the involved players (TR 701.3.b). A Game Loss or more severe penalty is reported to the head judge (TR 701.3.a). If the call, including the ruling and correction, took more than a minute, the players are given a time extension (TR 412.2 and TR 701.2.a.1). A player who disagrees can appeal to the head judge, but only after a full ruling has been made, and an appeal of a head judge's or appeals judge's ruling is not allowed (TR 413.1, TR 413.3 and TR 413.4). At high OPL the head judge may designate floor judges as appeals judges (TR 413.2).
Judges make mistakes too. A judge who recognizes their own mistake should acknowledge and correct it if possible, the head judge can downgrade a penalty that came from an official's mistake, and if a player acted on information given in error by an official, the head judge can rewind the game even if there were no other errors (TR 701.1.m, TR 701.1.n and TR 701.1.o). The head judge is the final authority on the rules, and only the head judge may issue penalties outside the guidelines (TR 204.4 and TR 701.1.k).