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...