A Partner is a Player who has formed a Pact with another Player. A Partner is essential to surviving the Reapers' Game because a pact is required to use the Psychs in order to defeat the Noise. At the start of every Game, the Players are required to find another partner as soon as possible or else they risk being killed by the Noise.
If a Partner is killed by the Noise, the surviving Partner only has around seven minutes to live until they are erased from existence. It is never fully explained why this is, although one possibility is that the Reapers are humane enough to let the surviving Player grieve the loss of their partner, as well as say goodbye to the other Players they may have befriended in their final moments.
In NEO: The World Ends with You, the concept of partners is very different as the Players are now categorized into teams. If a team member is erased, the group keeps playing. However, if the assigned team leader is erased, it seems the team as a whole is erased.