When you shoot another player in Fallout 76, you basically initiate PvP. The other player can run away or ignore you (your attacks inflict 50% damage, half your normal damage rate), but if they shoot you, you will both do full damage. If you kill the other player, you can loot their junk/scrap, but you cannot steal their weapons or caps. After killing another player, you will be marked in red as a murderer to everyone else on the server. You will also reach "Wanted" status. If the player you just killed comes back and gets a revenge kill on you, they will be able to loot double the junk/scrap off your body. To do PvP without consequence, play one of the following modes:

Hunter/Hunted mode: To join Hunter/Hunted mode, you must be at least Level 5. Then, go to the Radio Tab on your Pipboy, and select the "Hunter/Hunted" station. You will be transferred into a lobby to wait for a session to begin. A minimum of four players are required. The goal of Hunter/Hunted mode is to track down and kill another PC (player character) target. While you are hunting them, another player is hunting you. If you kill your player target, you will receive their target. Continue until you are the last player standing. There are special rewards for winning, or just doing well in this mode. Note: Once you join a Hunter/Hunted lobby, you can continue to complete quests, explore the map, and do whatever desired. You do not have to stay in the lobby.

The Battle That Never Was mode: This is a Team Deathmatch mode and can be started at Prickett's Fort. When you arrive, an event will trigger called "Event: The Battle That Never Was". You can choose your side (Union or Confederate), and fight other players that arrive in the area. Whichever team gets the most kills wins the match.