There are two primary ways to connect to i6: through SSH, or through SFTP.  Both are topics worthy of study on their own, but what follows is the barebones basics to get you up and running.

What is SSH?

SSH is a means for you to connect to i6 in the context of your i6 account and execute commands on i6 to do things like rename files, edit files, interact with a database, and execute programs on the server.

What is SFTP?

SFTP is a means for you to copy files to and from i6.

What is the difference?

If you suspect that there is some overlap to functionality here, you are correct.  For example, you can rename files by connecting via SSH and issuing a "mv" command, or you can connect via SFTP and (depending on the program), right click a file and select Rename.  Generally, SFTP is more limited than what SSH can do.  SFTP is strictly for transferring files from location A to location B, usually from your local computer, to your home/web directory on i6, whereas you can use your SSH session to issue the commands necessary to write and compile programs (just an example).

How do I SSH?

Windows: Download PuTTY.  Execute PuTTY and set the server to i6.cims.nyu.edu then connect.
Mac & Linux: Use the built-in Terminal.  If you can't find it, use the search function to look for "Terminal".  Once open, type "ssh NETID@i6.cims.nyu.edu".  Do note that you DO need to specify your NetID in this command, in lower case letters.  If you don't, ssh attempts to use your local computer's currently logged in username.

How do I SFTP?

Windows: Download WinSCP.
Mac: Download Fetch.
Linux: Consult your distributions documentation --- you may already have a GUI client installed.

*NOTE: the above are software recommendations.  They are by no means the only options available.