Competitive REL Tournament Primer
The Magic Tournament Rules state that players are responsible for being familiar with the rules contained within that document. Not knowing those rules may result in you having an unhappy experience at your first Competitive REL tournament. There are a lot of rules in the MTR, and most of them aren't that important to you, so this primer is a condensed version of what you really need to know before playing in your first Competitive tournament.
Additional information is available in the footnotes and external links, and a simplified printable PDF is available here. If you want to read the entire MTR, you can do so here.
As such, the atmosphere is a little more adversarial. Most players will of course still be friendly, and discussion is encouraged. But the primary goal is to win, learn from one's mistakes, and get better for next time; not to coach people through the game or have a social experience. The expectation at Competitive REL tournaments is that players are trying to win, and the enjoyment comes from the challenge of making good strategical plays and emerging triumphant against your opponent.
What to bring
Yourself, on time. The tournament will not wait for you if you're running late or unprepared. If you do get there late, you can still enter, but you'll take a loss in round 1 and only start playing in round 2. If you're late to your table for any of the later rounds, you may receive a penalty. If you think you're going to be late due to some important task (e.g. going to the bathroom) or circumstances outside your control, notify a judge in advance.
If it's a constructed tournament, you will need a decklist prepared in advance, which is a list of all cards in your deck and sideboard. You can print one out from here.
You'll need a way to track your life total. A piece of paper and a pen is best. A boogie board is also acceptable. Dice are not allowed, as they're too easily knocked over. A method that displays a history of past life changes will be a huge help in resolving any confusion or disputes that arise over life totals.
Be prepared to play with new sleeves, or at least sleeves in good condition. Old sleeves with damaged corners or dirt on the back may result in a penalty for Marked Cards. Bring some extra sleeves in case one breaks during the course of the tournament.
If any cards in your deck may put counters on anything, you should bring some dice to represent those counters. Dice are also helpful for determining who takes the first turn. Similarly, if your deck may create any tokens, you should bring tokens with you. Tokens can be anything that's roughly card-shaped and isn't going to confuse your opponent. (If it's a limited tournament, you'll probably be able to find the appropriate tokens from other players who opened them.)
You can bring some sideboarding notes to refer to in between games. These must be quick to access, and should be printed or written out rather than online. You may only look at these during sideboarding in between games of a match, not while a game is in progress.
Players at Competitive tournaments are held to a higher standard of play than at FNM or casual play. You should communicate clearly enough everything you're doing, and ensure that you and your opponent are on the same page at all times. (In particular, always state what step or phase of the game it is when you're taking an action, and announce all life total changes.) Brief takebacks are allowed, but once the game has moved on and your opponent may have made a decision based on seeing your play, you may not change your decision retroactively.
Questions about things like life totals, floating mana, the name of a creature, etc. must be answered fully and honestly. It is however ok to refuse to answer or lie about things like what cards are in your hand or what you're planning to do on your next turn. Some things, like the power of a creature on the battlefield, you don't have to answer, but you can't lie about. For the details, see section 4.1 of the MTR, or ask a judge.
Triggered abilities, such as "at the beginning of your upkeep", and "whenever a creature enters the battlefield", are the responsibility of their controller. If you see your opponent miss a trigger, it's ok for you to not mention it. For everything else, if you think you saw your opponent make a mistake, you must call for a judge.
Any time you need to shuffle your deck (or a portion of it), make sure you shuffle thoroughly, and present the deck to your opponent for them to shuffle it as well. (Don't look at the faces of any cards while you're shuffling.)
If anything goes wrong in your match, or you have any questions, raise your hand in the air and call "judge!", and a judge will be over to help you shortly. Make sure you're loud enough that they can hear you; don't worry about bothering the other players, they're used to it. Leave your hand raised so that the judge can easily locate you.
If you disagree with the judge's answer, it's perfectly acceptable to politely express this to them. (Explaining why you disagree will be much more effective than simply insisting that you're right.) If the two of you aren't able to reach a consensus, you have the right to an appeal to the head judge of the event. Tell the floor judge that you're appealing their ruling, and they'll go get the head judge. (You don't need to worry about this being rude to the judge; appeals are a normal part of a tournament, judges are well aware that they can make mistakes, and no competent judge would get offended over such a thing.)
If a judge tells you that you're receiving a Warning, that means you committed some infraction that needs to be tracked. A Warning itself has no effect on your tournament, but if you receiving multiple Warnings, they may start being upgraded to a Game Loss.
If you want to read the text of a card, you're allowed to look it up on your phone in the Companion App, Scryfall, or another resource. Any usage of your phone during the match must be visible to your opponent. If you wish to view the Oracle text of a card privately, you must call a judge.
At the end of your match, please remember to report the result in the Companion App, or whatever software your tournament is using. Late or missing results can significantly delay the next round. (And if you're dropping from the event, please mark yourself as dropping. It's rude to your next round opponent if they get paired against you and then you aren't there.)
What not to do
Do not have extra cards in your deckbox that are not a part of your deck or sideboard. This may result in you receiving a Game Loss.
Magic has very strict rules about what agreements you can make regarding your prize support and draws or concessions in your game. Do not wager anything on the tournament, including things like "loser has to drive home". Do not offer to give your opponent any of your prizes in return for a concession. Do not flip a coin to pick a winner of your match, or reveal cards off the top of your library after the match has gone to time. If you want a comprehensive explanation of what is and isn't allowed, there's one available here, but the short version is: as long as you just play Magic at the Magic tournament, you won't run afoul of any of these rules. If you have a specific question about what you're allowed to do, call for a judge and ask them about it out of earshot of your opponent.
If you're watching someone else's game, don't say anything that could potentially be interpreted as gameplay assistance. Do not talk about the game within earshot of the players. Don't make facial expressions or verbal exclamations based on seeing what card somebody just drew or a good play they just made. It's fine to talk to players in a match about things other than the game in progress, like "hey I'm gonna go get food, do you want anything?".
If you think you see a problem in a game you're watching, don't try to fix it yourself or inform the players about the details. Just say "hey, I think there might be a problem, could you pause your game for a moment?" Then call for a judge and explain the potential issue to them away from the table.
If you're participating in a Competitive REL draft, don't look at your picks unless given express permission to do so, and don't have multiple piles of cards in front of you that might be confusing. Keep all of your picks in a single face-down pile, and keep them clearly separate from any packs that you're handed.
Thank you to Robert Hinrichsen, Benjamin Valenzia, and Noah Rabin for each having previously made their own versions of this document, from which I drew inspiration. If you think this document is missing anything important, please contact me about it.