I have a laptop with a busted up graphics card. However I'm pretty sure that the rest of it works just fine. I was wondering if it was possible to use it as a home server and how I would do that given that I'm not able to see anything.
Absolutely; this is called a headless system. So long as you can get it to boot into a live distribution of Linux, you should be able to remotely configure the server via remote shell/terminal.
If you wish to use the current operating system, your only choice is to hope you can navigate without a graphical display. If you choose to use a live distribution of Linux, you have two choices.
You can boot it into "serial terminal mode" (offered even by the latest Ubuntu distribution), allowing you to connect a cable to the serial port (if the laptop has one), and communicate with the laptop that way. Secondly, you can also remotely configure the distro before booting, to automatically start networking services with a specified configuration, and then connect over the network.
Most modern laptops don't come with serial ("COM") ports anymore, so you might want to consider the latter approach.
If you need to modify the boot order (or see what keys to press to view a list of boot devices), check and see if you can download the manual. It should also show some of the BIOS screens to help you navigate through. Alternatively, knowing someone with the same model laptop (or at least with the same BIOS) may assist you with this process (also check Youtube for reviews, which usually cover the BIOS).
If your GPU simply fails to get an image to the built-in display, but can still produce an output on a different connector, this is trivial; simply connect a second monitor and continue using the machine normally.
If your motherboard can't even detect the presence of a video card, you're in trouble. The motherboard will give a beep code indicating no video card is found. Depending on the particular model and BIOS, it will either continue automatically, ask for a keystroke to resume booting (usually F1), or stop entirely. The behavior may be configured through the BIOS settings.
Assuming your laptop will function at all without a GPU, you'll have to work blindly. The process and difficulty differs per operating system. I have just tested the following procedure on my Windows 7 machine:
Note that when logging on to the remote desktop, you need a passworded administrator account.
If you want to install a fresh operating system, you might be able to automate the installation process up to a point where you can control a machine remotely. This article covers how to do this for Ubuntu, but similar procedures exist for many other OSes.
Though there are several solutions to installing a machine without a display, in my experience, they are very cumbersome and hard to troubleshoot. A GPU isn't easily replaced in a notebook, but you might want to look into a second video card you can plug into the USB port.
The problem is installing the initial operating system... without a working graphics card, you're pretty much stuck with whatever is already there. If you already have remote access software setup (ie: windows rdp or vnc), you're in a good place. If you don't, look for a way to install it, possibly via ssh or wmi, and you still might be okay.
The other good news is that we're starting to see some usb-based graphics cards on the market: walmart has one for about $60. The bad news is that they're not (yet) plug-n-play, and so setting one up usually already requires a working display... these are mainly intended for use with second or third monitors. You might get lucky, though, and find one that works with a linux live cd.