Right click on the external hard drive icon and select Get Info, a new window will open.
At the bottom of the new window, there is a Permissions section. Next to your user name there is an option with arrows. Click this and select Read & Write. This gives you permission to access the external drive to either write files to or delete files from the external drive.
If it already says Read & Write next to your user name, it could be the drive is formatted in FAT 32 or NTFS and the video file is too big to be copied over (FAT 32) or the NTFS file system isn't understood in Mac OS X. You would then have to use Disk Utility (Applications> Utilities> Disk Utility) to reformat the drive to GUID (in options) and Mac OS X (Extended). You can name your Hard Drive in the Space above Mac OS X (Extended).
Hope this helps