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