Erasure is the process of eliminating something from existence, an ability used by Players, Reapers, and Noise. It replaces "killing" throughout the Reapers' Game. Sanae Hanekoma's reports indicate that erasure only removes something from visible existence; their Soul remains in the Underground and can be reshaped.
In the original The World Ends with You, if a Player is defeated by Noise or by a Reaper, they are erased. In NEO: The World Ends with You a player defeated by another Player is not erased, and a player defeated by a Reaper is only erased if the Reaper chooses to do so after their defeat using a separate ability. It is also possible that Players defeated by Noise in NEO are not erased; although the game ends if the player's team is defeated by Noise, no other team speaks about having members erased by losing to Noise.
Reapers can erase Players if they pose an immediate threat to the running of the Reapers' Game. In the original, all Players are erased if no player completes a given day's mission; in NEO, it is not necessary to complete missions, but the team with the lowest score is erased at the end of the week.
If a player defeats a Noise, it is erased. Since Noise cannot have Souls, they are completely destroyed if erased. If a player defeats another Player or Reaper, they are not usually erased, but in TWEWY it is implied that they can be; in NEO they cannot be.
In TWEWY a Player who fails to make a Pact will be erased if they don't have a partner within a certain given amount of time. In addition, a player who has made a Pact is erased seven minutes after their Partner is erased. In NEO, it appears that Players can operate outside teams for a longer period of time, although they will be unable to score points without a team. It is possible that Players on a team are erased if the leader of that team is erased, although this is left ambiguous in the game.
It's possible to save a person shortly after their erasure by transferring their Soul into a Pin. This can be taken a step further by making the Pin into a Noise. This procedure can only be undertaken by Reapers or Angels, as they are the only ones who seem to be able to transfer Soul and create Noise.
Exorcism is a process introduced in NEO: The World Ends with You that is similar to erasure, albeit much more potent.
When someone is exorcised, not only are they completely removed from existence, but anyone who knew the person will have no immediate recollection of them. Even if one were to go back in time, the person exorcised would still be gone and no one would know who they are. However, those who forget about the person after their exorcism may have their memory jogged when reminded. This can range from completely remembering said person to only having a faint recollection of them.