Go to System Preferences and open Time Machine. In Time Machine, click Select Backup Disk and select your time machine disk there. Then click on Time Machine icon on the menu bar at the top left to WiFi etc icon, click Backup Now button. This will backup the entire Mac's hard drive and that will include your Photo Library. This is the recommended method.
You can also manually copy the iPhoto or Photos Library to your hard drive. To do that, connect the external hard drive to your Mac. This will recognize the external hard drive and it will appear in desktop. Now, in Finder, click on Go menu and select Home. This will open your Home folder. Open Pictures folder and you will find your iPhoto or Photo Libraries there. Just click and drag them to the external hard drive's icon on the desktop or external hard drive. This will locally copy them to external hard drive.
You can drag and drop any files to external hard drive this way and a copy will be saved on it.
Or just select your files, and press Command and C on your keyboard. Then open the external hard drive icon from desktop and press Command V and this will copy and paste the files there.
After copying the files to external hard drive, just compare the size of both copies of internal hard drive and external hard drive. To compare size, select the file (for eg : select iPhoto/Photos Library) and press Command I and this will bring Get Info window. Look at the file size. Like the same way, check the copied file size from external hard drive. If they are same, then you can delete them from internal hard drive. Double make sure you have the copy of your files saved on external hard drive before deleting them from your Mac.
Make sure to empty the trash to get the free space from deleted files.
If you have any difficulty on following the steps above, please let me know.