I am assuming you're trying to install magit though Melpa, so you probably downloaded the latest magit which is 2.3, and requires at least Emacs 24.4. If you don't have root access to the machine and can't upgrade any package, you'll have to fall back to the previous major version (1.4.2) which requires Emacs 23.2.
You can start by checking the README file of the 1.4.2 tree of magit at github, which has some basic information, but the real instructions are available at V1 documentation, which has some instruction for installing older versions.
Be aware that the instructions include some code you should put into your emacs configuration files, which is usually
~/.emacs.d/init.el. Since you mentioned you're an Emacs newbie you might not know how to place those lines into your configuration, so I recommend you to read a bit about it, it will be very useful for your Emacs life. Emacs manual and EmacsWiki can be good starting points. Some users use git to keep track of their configuration, so they can deploy their customized Emacs on almost any computer they need to work.
On the matter of lack of root access, one point to consider is that some distros are still using Emacs 24.3 and even with root access you won't be able to update, this is the case for Ubuntu 14.04 LTS and derivates. So even if you had admin access to the machine you'd still need to download and install Emacs from a secondary source other than your distro's repository, which in the end might not need root access (local install).