SSH stands for "Secure Shell", and it basically allows you to connect to your computer remotely. Here's a screenshot of sshing into Ubuntu from a Windows computer:

Using the SSH Server

First, we need to install the ssh server with:

sudo apt-get install openssh-server

To check that it's running, type:

sudo service ssh status

(to exit from the command, type 'q')
Make sure to unblock SSH on your firewall. If you're using Ubuntu with ufw:

sudo ufw allow ssh

Now that we've setup the server, we need to connect to it with a client.

Using the SSH Client

Install the SSH client if it isn't already there:

sudo apt-get install openssh-client

Now you can connect to the server:

ssh <user>@<ip address> -p <port>

<user> should be replaced with the user you want to login as, <ip address> should be the ip address of the computer or the hostname, and <port> should be the SSH port. You can find your ip address if you don't know it already with ip address (look for inet). The -p <port> part can be omitted if you set the SSH server to listen on the default port (22)
If it asks about the authenticity of the host type yes.
Now, it'll ask for your password.
If all goes well then you'll be connected to your remote computer!
Note that if you want to run graphical applications you'll need to have an X server installed.

Other operating systems

You can find how to install an SSH server on Windows here and how to do it on MacOS here. You can connect to SSH with the steps here.
If you have a chromebook, you can connect to ssh with the steps here. You can't start an ssh server without developer mode.