By default, the root folder on the new drive will be writable only by root.
There are a number of ways you can deal with this. Assuming you are hot-plugging it, and it is auto-mounting itself as /media/disk, any of the following should work (do this with the drive mounted) --
To make it writable by anyone:
sudo chmod go+w /media/disk
To change the ownership, making it writable by your account only:
sudo chown myaccountname:mygroupname /media/disk
To create a subfolder that is writable by your account, while leaving the root folder of the drive writable only by root:
sudo mkdir /media/disk/myfolder
sudo chown myaccountname:mygroupname /media/disk/myfolder
There's probably a way to do this through the GUI as well, but I've never bothered to figure out how since I'm comfortable with the CLI...
If the world isn't making sense to you, you're either drinking too much or not drinking enough.