SSH
General
Secure Shell (ssh) is a commandline-tool for logging into a different computer over some network (e.g. the internet) and for executing commands on that machine, as if one would be sitting there instead of the own computer. So you use ssh to build a connection to the other computer and can then interact with it, using it's shell. It is commonly used to login to the login nodes of a supercomputer.
Access
OpenSSH is the standard ssh client on Linux and Mac and it is freely available for everyone. On Windows you can use Putty or the GitBash (coming with ssh) which is also free.
Sometimes this access is restricted, so you can only connect, when you are within the university/facility and its network. To still access the Login Nodes externally, one can 'pretend to be inside the network' by using a Virtual Private Network (VPN).
Usage
Logging in with OpenSSH on a Linux Cluster is done with:
$ ssh -l <login> <cluster>
Here <login> is your username and <cluster> is one of the login nodes of the system, you are trying to connect to. At the first login you will be asked to verify the authenticity of the host. If the shown host is correct, enter yes. After pressing RETURN you will be prompted to enter your password for the provided username.
If you need to start graphical applications you need to enable X11 forwarding/X11 tunneling by your ssh client.
For OpenSSH this is done by giving it the command line option -X (or -Y if the previous did not work):
$ ssh -X -l <login> <cluster>
This might or might not work depending on the your operating system, because it requires an X11-Server running on your local machine, which is not available by default on Mac and Windows. To utilize graphical tools anyway, you might want to look into FastX, which provides the necessary functionality for Mac and Windows.
For more security and ease of use you should consider setting up authentication via ssh keys.