External hard drive files not viewable from Macbook air

I have a mid 2009 Macbook air, and recently purchased a 1TB Seegate external hard drive. I reformatted the drive for Mac and backed up my files but cannot see the folders, to confirm that all my pics are there. I am a new Mac owner and don't know a lot about how to navigate the drives. On Windows I used Windows Explorer to see the folders but on my Macbook air, I have not found an equivalent tool to see the folders and pics.
(I am a Canadian visiting Spain and taking pics daily and my Mac hard drive is full and won't let me download any more pics, and my camera disk is almost most full. I can buy a new memory disk for the camera, but I just spent $140 on the external hard drive to avoid this.)
Thanks for any help you can offer.

On a Mac, you simply double-click on the drive's icon on your desktop to open the main directory and begin exploring.

If your boot drive is full, that is going to cause all kinds of other problems and puts you at high risk of data loss. You need to find some stuff on your boot drive you can get rid of, like NOW.

Can you explain better what you mean by "backed up your files"? If you used Time Machine, then the only folder on your backup would be called "backups." Time Machine will tell you (using the icon in the menu bar) when the last backup was done.

On your boot drive, where the pictures are depends on what program you are using to import them. If you are using iPhoto, they're in the iPhoto Library safe and sound. Just open iPhoto to view them.

Does the drive show up in the Finder at all (on the desktop)? Make sure you are in the Finder then go to the Finder menu in the upper left of the screen and choose Preferences. When the next window opens choose the General tab and make sure "External disks" is checked.

If the drive appears you should be able to drag and drop files from your internal drive to the external. Once you are sure they have been copied you can delete them from the internal drive. Keep two things in mind though:

1. Do not delete files unless you are sure what they are. If you didn't create the file ask before deleting it.
2. When you drag a folder to the trash you don't actually get the space back on the drive until you empty the trash.