If the standby server is promoted to master, will it work without any replica? If I have 1 master and 1 standby server, and my master server is down and the standby server is promoted to master. Does my new master work fine without a replica?
Does it act like a standalone server?
Definitely, Your new primary will act like a standalone. If your configuration file is good, you'll even be able to add the old-primary as a new secondary without any restart.

Just be sure your old primary is not running and modifying data or you'll loose some of it!


  If the primary server fails and the standby server becomes the new
  primary, and then the old primary restarts, you must have a mechanism
  for informing the old primary that it is no longer the primary. This
  is sometimes known as STONITH (Shoot The Other Node In The Head),
  which is necessary to avoid situations where both systems think they
  are the primary, which will lead to confusion and ultimately data
  loss.


You may want to read these documentation page :
https://www.postgresql.org/docs/9.1/static/warm-standby-failover.html