Disk space (web space) is the amount of data you can store on your server. Think of it like its the hard drive of your web hosting account. You can store all your files in here. Your files can then be accessible (or not accessible if you so wish) for people to download from your website.
How much disk space you need depends on how much and what you're going to be storing. If you only want to host a blog and have no real downloads then you don't need a lot of disk space, however, if you want to store video or music files or any large file types then you will need quite a bit of free disk space. For this reason Marble Host offers huge disk space allowance to give you more freedom.