What does the Broken pipe message mean in an SSH session?

It's possible that your server closes connections that are idle for too long. You can update either your client (ServerAliveInterval) or your server (ClientAliveInterval)

 ServerAliveInterval
         Sets a timeout interval in seconds after which if no data has
         been received from the server, ssh(1) will send a message through
         the encrypted channel to request a response from the server.  The
         default is 0, indicating that these messages will not be sent to
         the server.  This option applies to protocol version 2 only.

 ClientAliveInterval
         Sets a timeout interval in seconds after which if no data has
         been received from the client, sshd(8) will send a message
         through the encrypted channel to request a response from the
         client.  The default is 0, indicating that these messages will
         not be sent to the client.  This option applies to protocol
         version 2 only.

To update your server (and restart your sshd)

echo "ClientAliveInterval 60" | sudo tee -a /etc/ssh/sshd_config

Or client-side:

echo "ServerAliveInterval 60" >> ~/.ssh/config 

An alternative solution would be to use mosh - the mobile shell. In contrast to ssh it connects via UDP and supports roaming. You can start your session at home, suspend your laptop, take it to work / friends / wherever else you have internet, unsuspend your laptop and continue to work as if nothing has happened. It is especially useful if you are on a lousy internet connection: It shows instant feedback if your keystrokes don't reach the server and continuously tries to reestablish the connection.

Installation and setup are simple: It is now included in all current Linux (plus a few non-Linux) distributions and it coordinates the session initialization and authentication via a prior ssh connection. So if you are able to connect via ssh user@server you are very likely to be able to connect with mosh just by calling mosh user@server, if the mosh packages are installed on both ends.

The main reason for connection failures is that you have to reach the server on a UDP port (default range: 60000-61000) for mosh to work. So if the server is behind a firewall you are mostly out of luck if can't punch holes in it yourself (security implications).

If you want to have a longer connection period, in the client add:

echo 'ServerAliveInterval 30' | tee -a ~/.ssh/config
echo 'ServerAliveCountMax 1200' | tee -a ~/.ssh/config

ServerAliveCountMax by default this is set to 3. Therefore once the ServerAliveInterval has sent 3 small packs of info to your server it will then automatically log out. Setting it to 1200 means this process will have to occur at least 1200 times. In short you should be connected at least 30*1200 seconds (10 hours).